Solving Equations By Factoring 6-8
Zero Products Property -- a "friend" you can count on..."
For any two rational numbers a and b, if ab = 0, then either a = 0 or b = 0 or both equal zero
How does this help you?
Try solving
x2 + 10x + 24 = 0
Your first reaction is to subtract 24 from both sides you would get
x2 + 10x = -24 BUT NOW WHAT?????
It's a quadratic ( x2 term). You can't isolate x because there is another term with the x2 in it. THe Zero Products Property will help you solve quadratics.
Using this property we can solve quadratic equations by factoring the equation and setting each factor to zero and solving them.
x2 + 10x + 24 = 0
factor as usual-- using your rules from the other sections:
(x +6) (x +4) = 0
set each one to zero
x + 6 = 0 and x + 4 = 0
so x = -6 and x = -4
Plug them back-- and it works!!
IF you are already given it factored--just set each to zero
(5x +1)(x-7) = 0
Using the Zero Products Property we know either 5x - 1 must be equal to zero or x - 7 must equal zero... that's the only way that the product could be zero...
Set both equal to zero and solve.
if 5x + 1 = 0 then x = -1/5 ( its just a 2 step equation)
if x - 7 = 0 then x = 7
If you substitute these answers for x back in the original equation, they will both end up as 0 = 0 which is what you want. The left side equals the right side!!
In the second example, the equation was already factored.
WIth the first example, you saw you needed to first factor the equation ( if possible) and then set each factor to zero to solve.
MAKE SURE YOU MOVE EVERYTHING TO ONE SIDE OF THE EQUATION. YOU MUST HAVE ZERO ON ONE SIDE OF THE EQUATION TO USE THE ZERO PRODUCTS PROPERTY
x2 = 16
First move the 16 to the left side
x2 -16 = 0
Now... you have the difference of two squares
(x + 4)(x-4) = 0
Set each factor to zero and solve
x + 4 = 0 and x - 4 = 0
so x = -4 and x = 4
Why do you have to have zero on one side ?
NEW ALGEBRA TERM
root-- any solution that turns the equation into the value of zero is called a root of the polynomial, or a ZERO of the polynomial because when you are graphing the polynomial, the y value is zero at this point.
So if the directions say " FInd the roots of..." it just means get zero on one side of th quation, factor, set each factor to zero and solve.
Summary of solving quadratics
solutions = roots= eros of the polynomials
1. Get zero on one side of the equation
2. Factor
3. Set each piece ( factor) equal to zero
4. Solve as one or two step equations
5. Check by substituting in the ORIGINAL equation
Wednesday, January 12, 2011
Tuesday, January 11, 2011
Math 6 Honors (Period 6 and 7)
Congruent Figures 4-7
Two figures are congruent if they have the same size and same shape. If we could lift on of the figures and place it directly on top of the other... all three vertices would match up with the others. In the example in our book ( page 132) we have two congruent triangles ∆ABC and ∆XYZ A would fall on X, B would fall on Y and C would fall on Z. These matching vertices are called corresponding vertices. Angles at corresponding vertices are corresponding angles and the sides joining corresponding vertices are corresponding sides.
The book states Corresponding angles of congruent figures are congruent.
and
Corresponding sides of congruent figures are congruent.
In class we discussed how to abbreviate the above -- when dealing with triangles.
CPCTC
Corresponding PARTS of congruent triangles are congruent!!
When we name two congruent figures we list corresponding vertices in the same order.
∆ABC ≅ ∆XYZ or ∆CAB ≅ ∆ZXY or ∆BCA ≅ ∆YZX
we know that
∠A ≅ ∠X and ∠B ≅ ∠Y and ∠C ≅ ∠Z
and the segments ( which are denoted with a line (but w/o arrows)above each of the two letters
AB ≅ XY and BC ≅ YZ and CA ≅ ZX
If two figures are congruent, we can make the coincide -- occupy the same place-- by using one or more of the following basic rigid motions:
Translation or slide
Rotation
Reflection or flip or mirror
Check the book on page 133 for good examples of these three rigid motions... I like to think of Tetris moves!!
Two figures are congruent if they have the same size and same shape. If we could lift on of the figures and place it directly on top of the other... all three vertices would match up with the others. In the example in our book ( page 132) we have two congruent triangles ∆ABC and ∆XYZ A would fall on X, B would fall on Y and C would fall on Z. These matching vertices are called corresponding vertices. Angles at corresponding vertices are corresponding angles and the sides joining corresponding vertices are corresponding sides.
The book states Corresponding angles of congruent figures are congruent.
and
Corresponding sides of congruent figures are congruent.
In class we discussed how to abbreviate the above -- when dealing with triangles.
CPCTC
Corresponding PARTS of congruent triangles are congruent!!
When we name two congruent figures we list corresponding vertices in the same order.
∆ABC ≅ ∆XYZ or ∆CAB ≅ ∆ZXY or ∆BCA ≅ ∆YZX
we know that
∠A ≅ ∠X and ∠B ≅ ∠Y and ∠C ≅ ∠Z
and the segments ( which are denoted with a line (but w/o arrows)above each of the two letters
AB ≅ XY and BC ≅ YZ and CA ≅ ZX
If two figures are congruent, we can make the coincide -- occupy the same place-- by using one or more of the following basic rigid motions:
Translation or slide
Rotation
Reflection or flip or mirror
Check the book on page 133 for good examples of these three rigid motions... I like to think of Tetris moves!!
Pre Algebra (Period 2 & 4)
Comparing and Ordering Fractions
5-1
HOW TO FIND THE LEAST COMMON MULTIPLE:
LCM = Smallest number that your numbers can go into.
Just like GCF, let's look at the letters backwards to understand it!
Multiple = each number given in the problem must go into this number (example for multiples of 2: 2, 4, 6, 8.... multiples of 3: 3, 6, 9, 12... multiples of 5: 5, 10, 15, 20...)
Common = must be a number that ALL THE NUMBERS go into
Least = must be the SMALLEST number that ALL THE NUMBERS go into
There are the same ways to find it as the GCF:
1) List multiples of each number and circle the smallest one that is common to all the numbers ---> most of the time takes WAY TOO long!!
2) Circle every factor in the prime factorizations of each number that is different and multiply
3) List the EXPANDED FORM prime factorizations in a table and bring down ONE OF EACH COLUMN. Then multiply. (or you can do this with exponential form but you need to bring down the HIGHEST POWER of each column).
4) Using the BOX method from class create an L from the left side and the bottom row of relatively prime factors. It is their product.
THE DIFFERENCE BETWEEN GCF AND LCM:
For the GCF, you need the LEAST POWER of only the COMMON FACTORS.
For the LCM, you need the GREATEST POWER of EVERY FACTOR.
WHY DO WE NEED EVERY FACTOR THIS TIME?
Because it's a multiple of all your numbers!
Multiples start with each number, so all the factors that make up each number have to be in this common multiple of all the numbers.
For example, say we're finding the LCM of 12 and 15, that multiple must be a multiple of 12: 12, 24, etc.
AND 15: 15, 30, etc.
So the COMMON multiple must include 12 (2x2x3) AND 15 (3x5).
The LCM must have two 2s and one 3 or 12 won't go into it.
It must also have that same 3 that 12 needs and one 5 or it won't be a multiple of 15.
EXAMPLE: 54 and 36
LIST MULTIPLES OF EACH NUMBER UNTIL YOU SEE ONE THAT BOTH OF THEM GO INTO (the LEAST MULTIPLE that is COMMON to both)
36 = 36, 72, 108, 144 ...
54 = 54, 108
This more difficult than the listing method for GCF for 2 reasons: You don't know where to stop as you least the first number...multiples go on forever! and the numbers get big very fast because they're MULTIPLES, not factors!
PRIME FACTORIZATION METHOD
54 = 2x3x3x3
36 = 2x2x3x3
The LCM will be one of each factor of each number (but don't double count a factor that is common to both numbers...once you have it in your LCM, check it off in the other number if it's common...you don't need that factor again)
LCM = 2x2x3x3x3 = 108
Let's look at this calculation and think about why it works:
Why does it need TWO 2s?
Although 54 only needs ONE, 36 needs TWO or 36 won't go into the LCM.
Try it putting in only ONE 2: 2x3x3x3 = 54. Does 36 go into 54? NO
Why does the LCM need THREE 3s? Although 36 needs only TWO 3s, 54 needs THREE. Try it putting in only TWO: 2x2x3x3 = 36.
36 is TOO SMALL to even be a MULTIPLE of 54!
WHAT IF BY MISTAKE YOU "DOUBLE" UP FACTORS AND USE ALL OF THEM?
For the GCF, you would have got a number way too big to be a factor of the numbers. For the LCM, you will STILL GET A COMMON MULTIPLE! But it WON'T BE THE LEAST!
In fact if you double up, generally you'll get a really big number and the bigger the number is, the harder it is to use.
For example, if you use all the factors of both 36 and 54, you're just multiplying 36 x 54 = ????
1944!!!!
Would you rather use 108 or 1944???
LCM WITH A COLUMN APPROACH: You place the prime factorization for each number in columns like we did for the GCF, matching the factors in each column. If a factor doesn't match, it gets a separate column.
YOU'LL JUST TAKE ONE OF EACH COLUMN AND MULTIPLY! SAME EXAMPLE: 54, 36
36 = 2 x 2 x 3 x 3
54 = 2 x 3 x 3 x 3
LCM = 2 x 2 x 3 x 3 x 3
= 108
You can do it in exponential form, too.
If you do it in exponential form, you take the HIGHEST POWER of each column!
Works with variables the exact same way!
Take the HIGHEST power of EVERY variable (not just the ones that are common like the GCF)
YOU CAN FIND BOTH THE GCF AND THE LCM IN THIS SAME COLUMN FORMAT!
CHECKING THE LCM TO MAKE SURE IT WORKS:
Ordering or comparing fractions:(last method)
4: Make them into decimals because DECIMALS = FRACTION posers!
(or decimals are just fraction wannabes!)
Today, we will change fractions to decimals and decimals to fractions.
How to change a fraction to a decimal 1.
Divide (ALWAYS WORKS!)
EXAMPLE: 3/4 = 3 divided by 4 = .75
If the quotient starts repeating, then put a bar over the number(s) that repeat. OR
2. Use equivalent fractions (SOMETIMES WORKS!)
Works if the denominator can be easily made into a power of 10
SAME EXAMPLE: but this time you will multiply by 25/25 to get 75/100 = .75
3. MEMORY! Some equivalencies you should just know! EXAMPLE: 1/2 = .5
IF IT'S A MIXED NUMBER, JUST ADD THE WHOLE NUMBER AT THE END!
EXAMPLE: 8 3/4
For the fraction: 3 divided by 4 = .75
Add the whole number: 8.75
IF THE MIXED NUMBER OR FRACTION IS NEGATIVE, SO IS THE DECIMAL!
CHANGING TERMINATING DECIMALS TO FRACTIONS: EASY!!!
Read it, Write it, Simplify!
EXAMPLE:
Change .24 to a fraction
1) READ IT: 24 hundredths
2) WRITE IT: 24/100
3) SIMPLIFY: 24/100 = 6/25
EXAMPLE with whole number:
Change 7.24 to a fraction
The 7 is the whole number in the mixed number so you just put the 7 at the end
1) READ IT: 24 hundredths
2) WRITE IT: 24/100
3) SIMPLIFY: 24/100 = 6/25
4) 7 6/25
HOW TO CHANGE REPEATING DECIMALS TO FRACTIONS---> we will do that 2nd semester
HOW TO FIND THE LEAST COMMON MULTIPLE:
LCM = Smallest number that your numbers can go into.
Just like GCF, let's look at the letters backwards to understand it!
Multiple = each number given in the problem must go into this number (example for multiples of 2: 2, 4, 6, 8.... multiples of 3: 3, 6, 9, 12... multiples of 5: 5, 10, 15, 20...)
Common = must be a number that ALL THE NUMBERS go into
Least = must be the SMALLEST number that ALL THE NUMBERS go into
There are the same ways to find it as the GCF:
1) List multiples of each number and circle the smallest one that is common to all the numbers ---> most of the time takes WAY TOO long!!
2) Circle every factor in the prime factorizations of each number that is different and multiply
3) List the EXPANDED FORM prime factorizations in a table and bring down ONE OF EACH COLUMN. Then multiply. (or you can do this with exponential form but you need to bring down the HIGHEST POWER of each column).
4) Using the BOX method from class create an L from the left side and the bottom row of relatively prime factors. It is their product.
THE DIFFERENCE BETWEEN GCF AND LCM:
For the GCF, you need the LEAST POWER of only the COMMON FACTORS.
For the LCM, you need the GREATEST POWER of EVERY FACTOR.
WHY DO WE NEED EVERY FACTOR THIS TIME?
Because it's a multiple of all your numbers!
Multiples start with each number, so all the factors that make up each number have to be in this common multiple of all the numbers.
For example, say we're finding the LCM of 12 and 15, that multiple must be a multiple of 12: 12, 24, etc.
AND 15: 15, 30, etc.
So the COMMON multiple must include 12 (2x2x3) AND 15 (3x5).
The LCM must have two 2s and one 3 or 12 won't go into it.
It must also have that same 3 that 12 needs and one 5 or it won't be a multiple of 15.
EXAMPLE: 54 and 36
LIST MULTIPLES OF EACH NUMBER UNTIL YOU SEE ONE THAT BOTH OF THEM GO INTO (the LEAST MULTIPLE that is COMMON to both)
36 = 36, 72, 108, 144 ...
54 = 54, 108
This more difficult than the listing method for GCF for 2 reasons: You don't know where to stop as you least the first number...multiples go on forever! and the numbers get big very fast because they're MULTIPLES, not factors!
PRIME FACTORIZATION METHOD
54 = 2x3x3x3
36 = 2x2x3x3
The LCM will be one of each factor of each number (but don't double count a factor that is common to both numbers...once you have it in your LCM, check it off in the other number if it's common...you don't need that factor again)
LCM = 2x2x3x3x3 = 108
Let's look at this calculation and think about why it works:
Why does it need TWO 2s?
Although 54 only needs ONE, 36 needs TWO or 36 won't go into the LCM.
Try it putting in only ONE 2: 2x3x3x3 = 54. Does 36 go into 54? NO
Why does the LCM need THREE 3s? Although 36 needs only TWO 3s, 54 needs THREE. Try it putting in only TWO: 2x2x3x3 = 36.
36 is TOO SMALL to even be a MULTIPLE of 54!
WHAT IF BY MISTAKE YOU "DOUBLE" UP FACTORS AND USE ALL OF THEM?
For the GCF, you would have got a number way too big to be a factor of the numbers. For the LCM, you will STILL GET A COMMON MULTIPLE! But it WON'T BE THE LEAST!
In fact if you double up, generally you'll get a really big number and the bigger the number is, the harder it is to use.
For example, if you use all the factors of both 36 and 54, you're just multiplying 36 x 54 = ????
1944!!!!
Would you rather use 108 or 1944???
LCM WITH A COLUMN APPROACH: You place the prime factorization for each number in columns like we did for the GCF, matching the factors in each column. If a factor doesn't match, it gets a separate column.
YOU'LL JUST TAKE ONE OF EACH COLUMN AND MULTIPLY! SAME EXAMPLE: 54, 36
36 = 2 x 2 x 3 x 3
54 = 2 x 3 x 3 x 3
LCM = 2 x 2 x 3 x 3 x 3
= 108
You can do it in exponential form, too.
If you do it in exponential form, you take the HIGHEST POWER of each column!
Works with variables the exact same way!
Take the HIGHEST power of EVERY variable (not just the ones that are common like the GCF)
YOU CAN FIND BOTH THE GCF AND THE LCM IN THIS SAME COLUMN FORMAT!
CHECKING THE LCM TO MAKE SURE IT WORKS:
Ordering or comparing fractions:(last method)
4: Make them into decimals because DECIMALS = FRACTION posers!
(or decimals are just fraction wannabes!)
Today, we will change fractions to decimals and decimals to fractions.
How to change a fraction to a decimal 1.
Divide (ALWAYS WORKS!)
EXAMPLE: 3/4 = 3 divided by 4 = .75
If the quotient starts repeating, then put a bar over the number(s) that repeat. OR
2. Use equivalent fractions (SOMETIMES WORKS!)
Works if the denominator can be easily made into a power of 10
SAME EXAMPLE: but this time you will multiply by 25/25 to get 75/100 = .75
3. MEMORY! Some equivalencies you should just know! EXAMPLE: 1/2 = .5
IF IT'S A MIXED NUMBER, JUST ADD THE WHOLE NUMBER AT THE END!
EXAMPLE: 8 3/4
For the fraction: 3 divided by 4 = .75
Add the whole number: 8.75
IF THE MIXED NUMBER OR FRACTION IS NEGATIVE, SO IS THE DECIMAL!
CHANGING TERMINATING DECIMALS TO FRACTIONS: EASY!!!
Read it, Write it, Simplify!
EXAMPLE:
Change .24 to a fraction
1) READ IT: 24 hundredths
2) WRITE IT: 24/100
3) SIMPLIFY: 24/100 = 6/25
EXAMPLE with whole number:
Change 7.24 to a fraction
The 7 is the whole number in the mixed number so you just put the 7 at the end
1) READ IT: 24 hundredths
2) WRITE IT: 24/100
3) SIMPLIFY: 24/100 = 6/25
4) 7 6/25
HOW TO CHANGE REPEATING DECIMALS TO FRACTIONS---> we will do that 2nd semester
Wednesday, January 5, 2011
Circles 4-6 continued
We continued our study of circles by examining irregular shapes and determined their perimeters.
To see each of the irregular shapes turned to page 131. The numbers we used in class were all different that those of 24-27, but use the shapes to help solve the following:
The circles in the diagrams are parts of circles and the angles are right angles. We found the perimeter of each figure.
The first figure was a semicircle ( see #24 with a diameter of 4).
We noticed that we needed to start with
C =∏d but then we only need half of that
so
∏d/2 or 4∏/2 = 2∏
Using ∏≈ 3.14
we found
≈3.14(2) = 6.26
BUT.. that only was the upper part we needed to add the diameter of 4 to make sure we had all we needed in our perimeter.
6.28 + 4 = 10.28 units... but then we needed to round to 3 digits-- according to our textbook so
10.3 units would be a good approximation for the first perimeter.
Our 2nd irregular shape was a quarter of a circle with a radius of 6
Again, look to our textbook, page 131 # 25 for the shape. Use 6 as the radius.
This time
C = 2∏r
BUT... we only need 1/4 so
(2⋅∏⋅6)/4 = 12∏/4 = 3∏
≈ 3.14(3) = 9.42
BUT... wait.. we aren't finished... we have to sides of this quarter circle that we need to include in our perimeter.
so the perimeter is approximately 9.42 + 6 + 6 = 9.42 + 12
≈ 21.42 which round to ≈ 21.4 units
The next irregular shape looks like something from Griffith Park observatory. Make sure to use the diagram for # 26 but use a radius of 2 as we did in class.
C = 2∏r
but you notice we only need half of the full circle so
2∏r/2 or just ∏r
Now substitute int he radius-- which is also 2
2∏ ≈ 2(3.14) = 6.28
But... we still need the bottom perimeter
so this shape ≈ 6.28 + 2 + 4 + 2 or 6.28 + 8
≈14.28
≈14.3 units
Period 7 said the next shape ( # 27 in our textbook) looks like a bandaid... What do you think?
(At least.. with the way I drew it.. having a radius of 10)
Again we need the formula
C = 2∏r
This time we realized we had two semicircles.. but that is one whole circles so we kept the formula and substituted in our radius of 10
C =2(10)∏
=20∏
≈20(3.14)= 62.8
Then we added the two sides of 10 and found the perimeter to be approximately
62.8 + 10 + 10
≈82.8 units
I described the last shape to be a teardrop. Make sure to check #28 in our textbook. I actually used the same radius as the book so that drawing is exactly what we did.
C =2∏r
C =2⋅6⋅∏ = 12∏
But... we only need 3/4 of the circle so what is 3/4 of 12... in class everyone knew it was 9 so
3/4 of 12∏ is 9∏
≈9(3.14) = 28.26
But then we need to make sure we include the two sides of 6 each
≈28.26 + 6 + 6 = 28.26 + 12
≈40.26 units
and rounding to three digits
≈ 40.3 units.
We continued our study of circles by examining irregular shapes and determined their perimeters.
To see each of the irregular shapes turned to page 131. The numbers we used in class were all different that those of 24-27, but use the shapes to help solve the following:
The circles in the diagrams are parts of circles and the angles are right angles. We found the perimeter of each figure.
The first figure was a semicircle ( see #24 with a diameter of 4).
We noticed that we needed to start with
C =∏d but then we only need half of that
so
∏d/2 or 4∏/2 = 2∏
Using ∏≈ 3.14
we found
≈3.14(2) = 6.26
BUT.. that only was the upper part we needed to add the diameter of 4 to make sure we had all we needed in our perimeter.
6.28 + 4 = 10.28 units... but then we needed to round to 3 digits-- according to our textbook so
10.3 units would be a good approximation for the first perimeter.
Our 2nd irregular shape was a quarter of a circle with a radius of 6
Again, look to our textbook, page 131 # 25 for the shape. Use 6 as the radius.
This time
C = 2∏r
BUT... we only need 1/4 so
(2⋅∏⋅6)/4 = 12∏/4 = 3∏
≈ 3.14(3) = 9.42
BUT... wait.. we aren't finished... we have to sides of this quarter circle that we need to include in our perimeter.
so the perimeter is approximately 9.42 + 6 + 6 = 9.42 + 12
≈ 21.42 which round to ≈ 21.4 units
The next irregular shape looks like something from Griffith Park observatory. Make sure to use the diagram for # 26 but use a radius of 2 as we did in class.
C = 2∏r
but you notice we only need half of the full circle so
2∏r/2 or just ∏r
Now substitute int he radius-- which is also 2
2∏ ≈ 2(3.14) = 6.28
But... we still need the bottom perimeter
so this shape ≈ 6.28 + 2 + 4 + 2 or 6.28 + 8
≈14.28
≈14.3 units
Period 7 said the next shape ( # 27 in our textbook) looks like a bandaid... What do you think?
(At least.. with the way I drew it.. having a radius of 10)
Again we need the formula
C = 2∏r
This time we realized we had two semicircles.. but that is one whole circles so we kept the formula and substituted in our radius of 10
C =2(10)∏
=20∏
≈20(3.14)= 62.8
Then we added the two sides of 10 and found the perimeter to be approximately
62.8 + 10 + 10
≈82.8 units
I described the last shape to be a teardrop. Make sure to check #28 in our textbook. I actually used the same radius as the book so that drawing is exactly what we did.
C =2∏r
C =2⋅6⋅∏ = 12∏
But... we only need 3/4 of the circle so what is 3/4 of 12... in class everyone knew it was 9 so
3/4 of 12∏ is 9∏
≈9(3.14) = 28.26
But then we need to make sure we include the two sides of 6 each
≈28.26 + 6 + 6 = 28.26 + 12
≈40.26 units
and rounding to three digits
≈ 40.3 units.
Tuesday, January 4, 2011
Math 6 Honors (Period 6 and 7)
Circles 4-6
A circle is the set of all points in a plan at a given distance from a given point O (called the center).
A segment joining the center to a point on the circle is called a radius ( plural: radii) of the circle. All radii of a given circle have the same length and the length is called the radius of the circle.
A segment joining two points on a circle is called a chord... and a chord passing through the center is a diameter of the circle. the ends of the diameter divide the circle into two semicircles. The length of a diameter is called the diameter of the circle.
Two radii equal one diameter-- a fact we will use in the formulas below
The perimeter of a circle is called the circumference and the quotient
circumference ÷ diameter is the same for all circles--> regardless of size
This quotient is denoted by the Greek letter ∏ ( pronounced "pie")
No decimal gives ∏ exactly
No fraction gives ∏ exactly, either
A fairly good approximation is either 3.14 or 22/7
If we denote the circumference by C and the diameter by d we can write
C ÷ d = ∏
This formula can be put into several useful forms.
Let C = circumference d = diameter and r = radius
Then:
C = ∏d
d = C/∏
C = 2∏r
and
r = C/(2∏)
We tried a few examples.
Using ∏≈ 3.14 and rounding to three digits, as described by our textbook.
The diameter of a circle is 6 cm. Find the circumference.
WE are given d and are asked to find C.
WE use the formula
C = ∏d
C ≈ 3.14(6) = 18.84
C ≈ 18.8
So, the circumference is approximately 18.8 cm
The circumference of a circle is 20 feet. Find the radius.
To find the radius, use the formula
r = C/(2∏)
r = 20/2∏
Simplify first
r = 10/∏
r ≈ 10/3.14
r ≈ 3.1847
Since the third digit from the left is in the hundredths' place, round to the nearest hundredth.
r ≈ 3.18
The radius is approximately 3.18 feet
A polygon is inscribed in a circle if all of its vertices are on the circle. Check on the diagram in our textbook on page 129-- we added that to our notes as well.
Three noncollinear points (not on a line) determine one and only one circle that passes through the three given points.
A circle is the set of all points in a plan at a given distance from a given point O (called the center).
A segment joining the center to a point on the circle is called a radius ( plural: radii) of the circle. All radii of a given circle have the same length and the length is called the radius of the circle.
A segment joining two points on a circle is called a chord... and a chord passing through the center is a diameter of the circle. the ends of the diameter divide the circle into two semicircles. The length of a diameter is called the diameter of the circle.
Two radii equal one diameter-- a fact we will use in the formulas below
The perimeter of a circle is called the circumference and the quotient
circumference ÷ diameter is the same for all circles--> regardless of size
This quotient is denoted by the Greek letter ∏ ( pronounced "pie")
No decimal gives ∏ exactly
No fraction gives ∏ exactly, either
A fairly good approximation is either 3.14 or 22/7
If we denote the circumference by C and the diameter by d we can write
C ÷ d = ∏
This formula can be put into several useful forms.
Let C = circumference d = diameter and r = radius
Then:
C = ∏d
d = C/∏
C = 2∏r
and
r = C/(2∏)
We tried a few examples.
Using ∏≈ 3.14 and rounding to three digits, as described by our textbook.
The diameter of a circle is 6 cm. Find the circumference.
WE are given d and are asked to find C.
WE use the formula
C = ∏d
C ≈ 3.14(6) = 18.84
C ≈ 18.8
So, the circumference is approximately 18.8 cm
The circumference of a circle is 20 feet. Find the radius.
To find the radius, use the formula
r = C/(2∏)
r = 20/2∏
Simplify first
r = 10/∏
r ≈ 10/3.14
r ≈ 3.1847
Since the third digit from the left is in the hundredths' place, round to the nearest hundredth.
r ≈ 3.18
The radius is approximately 3.18 feet
A polygon is inscribed in a circle if all of its vertices are on the circle. Check on the diagram in our textbook on page 129-- we added that to our notes as well.
Three noncollinear points (not on a line) determine one and only one circle that passes through the three given points.
Tuesday, December 14, 2010
Math 6 Honors (Period 6 and 7)
Polygons 4-5
A polygon is a closed figure formed by joining segments ( the sides of the polygon) at their endpoints ( the vertices of the polygon). Polygons are named for the number of sides they have.
Triangle- 3 sides
Quadrilateral- 4 sides
Pentagon- 5 sides
Hexagon- 6 sides
Octagon- 8 sides
Decagon- 10 sides
A polygon is regular if ALL of its sides are congruent and ALL of its angles are congruent.
To name the polygon we name its consecutive vertices IN ORDER.
A diagonal of a polygon is a segment joining two nonconsecutive vertices.
Look at the quadrilateral on Page 123 and notice the two segments that represent the diagonals of the quadrilateral PQRS.
Certain quadrilaterals have special names.
A parallelogram has it opposite sides parallel and congruent.
A trapezoid has just one pair of parallel sides.
Certain parallelograms also have special names
A rhombus ( rhombii plural) has all it sides congruent.
A square has congruent sides and congruent angles
A rectangle has all its angles congruent.
Thus a square is a rectangle.. but a rectangle isn't necessarily a square!!
TH\he perimeter of a figure is the distance (think fence) around it. Thus the perimeter of a polygon is the sum of the lengths of its sides.
Always label your perimeters. If the figure provides a specific measure, such as meters (m), centimeters (cm), feet (ft), inches (in.)-- make sure to use that label.
If no unit of measure is given, always include "units"
A polygon is a closed figure formed by joining segments ( the sides of the polygon) at their endpoints ( the vertices of the polygon). Polygons are named for the number of sides they have.
Triangle- 3 sides
Quadrilateral- 4 sides
Pentagon- 5 sides
Hexagon- 6 sides
Octagon- 8 sides
Decagon- 10 sides
A polygon is regular if ALL of its sides are congruent and ALL of its angles are congruent.
To name the polygon we name its consecutive vertices IN ORDER.
A diagonal of a polygon is a segment joining two nonconsecutive vertices.
Look at the quadrilateral on Page 123 and notice the two segments that represent the diagonals of the quadrilateral PQRS.
Certain quadrilaterals have special names.
A parallelogram has it opposite sides parallel and congruent.
A trapezoid has just one pair of parallel sides.
Certain parallelograms also have special names
A rhombus ( rhombii plural) has all it sides congruent.
A square has congruent sides and congruent angles
A rectangle has all its angles congruent.
Thus a square is a rectangle.. but a rectangle isn't necessarily a square!!
TH\he perimeter of a figure is the distance (think fence) around it. Thus the perimeter of a polygon is the sum of the lengths of its sides.
Always label your perimeters. If the figure provides a specific measure, such as meters (m), centimeters (cm), feet (ft), inches (in.)-- make sure to use that label.
If no unit of measure is given, always include "units"
Algebra (Period 1)
Factoring by Group 6-6
First a review:
Checklist of how to factor thus far-->
1. Look for a GCF of all terms
2. Binomials - look for difference of two squares
both perfect squares - double hug - one positive, one negative - square roots of both terms
3. Trinomials - look for Trinomial Square (factors as a binomial squared)
first and last must be perfect squares - middle must be double the product of the two square roots
SINGLE hug - square roots of both terms - sign is middle sign
4. Trinomials - last sign positive - double hug with same sign as middle term - factors that multiply to last and add to middle
5. Trinomials - last sign negative - double hug with different signs, putting middle sign in first hug - factors that multiply to last and subtract to middle - middle sign will always be with the bigger factor
REMEMBER: FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!
(we're skipping 6-5 and then going back to it)
When you have 4 TERMS IN YOUR POLYNOMIAL!
You put the polynomial in 2 sets of 2 by using ( )
Then you factor out the GCF for each set of 2 terms individually
DOES THIS ALWAYS WORK FOR EVERY 4 TERM POLYNOMIAL?
Of course not!
But for this section of the math book, it will!
What happens if it doesn't work? The polynomial may just not be factorable!
MAKE SURE IT'S IN DESCENDING ORDER FIRST!!!!
EXAMPLE: 6x3 - 9x2 + 4x - 6
First notice there is NO GCF of all the terms!!
Factoring by grouping says if there is no GCF of the 4 terms, look and see if there is a GCF of just 2 terms at a time!!
Put ( ) around the first 2 terms and another ( ) around the 2nd set of terms.
(6x3 - 9x2) + (4x - 6)
Factor out the GCF from each set of two terms
3x2(2x - 3) + 2(2x - 3)
Look for a COMMON factor to factor out between the two sets
In this case its (2x - 3)
Pull out
(2x - 3)(3x + 2)
and check to make sure you cannot continue to factor!!
Try these:
x3 + x2 + 2x - 2
First... is there a GCF? No
okay
now set up in 2 groups of TWO
(x3 + x2) + (2x - 2)
x2 (x + 1) + 2 (x-1)
wait... they are NOT the same...
cannot be factored.. not factorable!!
2x2 - 4x + xz - 2z
Is there a GCF? NO
(2x2 - 4x) + (xz - 2z)
2x(x -2) + z(x-2)
(x-2)(2x + z)
24x3 + 27x2 - 8x - 9
Is there a GCF? NO
(24x3 + 27x2) + (-8x - 9)
3x2(8x +9) -1(8x + 9)
(8x + 9)(3x2 - 1)
c6 -c4 - c2 + 1
(c6 -c4) + (-c2 + 1)
c4(c2 -1) -1c2 -1)
Look carefully at that results.. why did the second term become -1?
(c2 -1)(c4 -1)
and ask yourself... are you finished factoring? ...
NO
I see The difference of Two Squares...
(c + 1)(c-1)(c2+1)(c2 -1)
Now are you finished?
No.. I still see the difference of Two Squares... bring everything down...
(c + 1)(c-1)(c2+1)(c + 1)(c-1)
and now write it in the correct order
(c2 + 1)(c + 1)(c + 1)(c-1)(c-1)
4y5 + 6y4 +6y3 +9y2
First thing-- Is there a GCF? YES
pull out a y2 and you are left with
y2(4y3 + 6y2 + 6y + 9)
Now put those 4 terms in 2 groups of two!!.. use brackets..
y2[(4y3 + 6y2) + 6y + 9)]
Look for a GCF in each of the hugs!!
y2[2y2(2y +3) + 3(2y+3)]
What do each of them have in common? What do they share? 2y + 3
when you put that in the first set of hugs... what's left?
y2(2y+3)(2y2+3)
First a review:
Checklist of how to factor thus far-->
1. Look for a GCF of all terms
2. Binomials - look for difference of two squares
both perfect squares - double hug - one positive, one negative - square roots of both terms
3. Trinomials - look for Trinomial Square (factors as a binomial squared)
first and last must be perfect squares - middle must be double the product of the two square roots
SINGLE hug - square roots of both terms - sign is middle sign
4. Trinomials - last sign positive - double hug with same sign as middle term - factors that multiply to last and add to middle
5. Trinomials - last sign negative - double hug with different signs, putting middle sign in first hug - factors that multiply to last and subtract to middle - middle sign will always be with the bigger factor
REMEMBER: FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!
(we're skipping 6-5 and then going back to it)
When you have 4 TERMS IN YOUR POLYNOMIAL!
You put the polynomial in 2 sets of 2 by using ( )
Then you factor out the GCF for each set of 2 terms individually
DOES THIS ALWAYS WORK FOR EVERY 4 TERM POLYNOMIAL?
Of course not!
But for this section of the math book, it will!
What happens if it doesn't work? The polynomial may just not be factorable!
MAKE SURE IT'S IN DESCENDING ORDER FIRST!!!!
EXAMPLE: 6x3 - 9x2 + 4x - 6
First notice there is NO GCF of all the terms!!
Factoring by grouping says if there is no GCF of the 4 terms, look and see if there is a GCF of just 2 terms at a time!!
Put ( ) around the first 2 terms and another ( ) around the 2nd set of terms.
(6x3 - 9x2) + (4x - 6)
Factor out the GCF from each set of two terms
3x2(2x - 3) + 2(2x - 3)
Look for a COMMON factor to factor out between the two sets
In this case its (2x - 3)
Pull out
(2x - 3)(3x + 2)
and check to make sure you cannot continue to factor!!
Try these:
x3 + x2 + 2x - 2
First... is there a GCF? No
okay
now set up in 2 groups of TWO
(x3 + x2) + (2x - 2)
x2 (x + 1) + 2 (x-1)
wait... they are NOT the same...
cannot be factored.. not factorable!!
2x2 - 4x + xz - 2z
Is there a GCF? NO
(2x2 - 4x) + (xz - 2z)
2x(x -2) + z(x-2)
(x-2)(2x + z)
24x3 + 27x2 - 8x - 9
Is there a GCF? NO
(24x3 + 27x2) + (-8x - 9)
3x2(8x +9) -1(8x + 9)
(8x + 9)(3x2 - 1)
c6 -c4 - c2 + 1
(c6 -c4) + (-c2 + 1)
c4(c2 -1) -1c2 -1)
Look carefully at that results.. why did the second term become -1?
(c2 -1)(c4 -1)
and ask yourself... are you finished factoring? ...
NO
I see The difference of Two Squares...
(c + 1)(c-1)(c2+1)(c2 -1)
Now are you finished?
No.. I still see the difference of Two Squares... bring everything down...
(c + 1)(c-1)(c2+1)(c + 1)(c-1)
and now write it in the correct order
(c2 + 1)(c + 1)(c + 1)(c-1)(c-1)
4y5 + 6y4 +6y3 +9y2
First thing-- Is there a GCF? YES
pull out a y2 and you are left with
y2(4y3 + 6y2 + 6y + 9)
Now put those 4 terms in 2 groups of two!!.. use brackets..
y2[(4y3 + 6y2) + 6y + 9)]
Look for a GCF in each of the hugs!!
y2[2y2(2y +3) + 3(2y+3)]
What do each of them have in common? What do they share? 2y + 3
when you put that in the first set of hugs... what's left?
y2(2y+3)(2y2+3)
Pre Algebra (Period 2 & 4)
Simplifying Fractions 4-4
Equivalent Fractions - Just multiply the numerator and the denominator by the same number and you will get an equivalent (equal) fraction to the one you started with.
GOLDEN RULE OF FRACTIONS = Do unto the numerator as you do unto the denominator
Simplifying fractions (your parents call this "reducing")
2 good ways:
(1) Just divide both the numerator and denominator by the GCF
(2) Another way: Rewrite the numerator and denominator in prime factorization form. Then simply cross out each common factor on the top and bottom
(they cross out because it's 1)
You'll be left with the simplified fraction every time!!!!
THE GCF METHOD:
One of the reasons we learn the GCF is because it's the FASTEST WAY TO SIMPLIFY FRACTIONS IN ONE STEP!!!
Just divide both the numerator and denominator by the GCF
THE PROBLEM WITH THE METHOD:
If you're not comfortable finding the GCF, you're pretty much sunk with this method! :(
THE BEST REASON TO USE THIS METHOD (other than it's a Calif. STAR Key Standard), it truly is the FASTEST :)
So imagine you have a "GCF Magical Voice" in your head...
The voice tells you the GCF of the numerator and the denominator...
You simply use that GCF to divide both the top and bottom of your fraction and you're done in one step!
THE PRIME FACTORIZATION METHOD:
This is sort of using the GCF "incognito" (in disguise)!
Rewrite the numerator and denominator in prime factorization form.
(Use a Factor Tree or Inverted Division to find the Prime Factorization if necessary).
Then simply cross out each common factor on the top and bottom.
(You're actually using the ID Property of Multiplication because each "crossout" is really a quotient of 1!)
You'll be left with the simplified fraction every time!!!!
If you actually multiplied together all your cross-outs, you'd get the GCF...
so you're using the GCF without even computing it!
THE PROBLEM WITH THIS METHOD:
You may think it's a lot of work
THE BEST REASON TO USE THIS METHOD: Although it takes time, everyone can do a Factor Tree or Inverted Division and create the Prime Factorization...
You'll never get the wrong answer with this one!
THE CROSS OUT METHOD:
You simply think of the first number that comes to your mind that "goz-into" both the numerator and the denominator and keep going until it's simplified.
If it's even, most people start with dividing it in half....and then in half again, etc.
This probably takes the longest, but in practice, most people use this method!
THE PROBLEM WITH THIS METHOD: You may think that a fraction is simplified, but you've missed a factor...this especially happens when the number is odd and you're always used to using 2 to divide the top and the bottom!
THE BEST REASON TO USE THIS METHOD: No one ever forgets how to do this method...it just comes naturally and there are no "precise" steps to do!
EXAMPLE: Simplify by each method: 36/ 54
GCF METHOD:
The GCF is 18:
36 ÷ 18 = 2
54 ÷ 18 = 3
PRIME FACTORIZATION METHOD:
36 = 2 x 2 x 3 x 3
54 = 2 x 3 x 3 x 3
Two of the 3s cross out and one of the 2s
You are now left with:
2/ 3
That's it!!!!!!!!!
CROSS OUT METHOD:
36 ÷ 2 = 18 ÷ 3 = 6 ÷ 3 = 2
54 ÷ 2 = 27 ÷ 3 = 9 ÷ 3 = 3
so 36/54 = 2/3
Do the same thing with variables!
Equivalent Fractions - Just multiply the numerator and the denominator by the same number and you will get an equivalent (equal) fraction to the one you started with.
GOLDEN RULE OF FRACTIONS = Do unto the numerator as you do unto the denominator
Simplifying fractions (your parents call this "reducing")
2 good ways:
(1) Just divide both the numerator and denominator by the GCF
(2) Another way: Rewrite the numerator and denominator in prime factorization form. Then simply cross out each common factor on the top and bottom
(they cross out because it's 1)
You'll be left with the simplified fraction every time!!!!
THE GCF METHOD:
One of the reasons we learn the GCF is because it's the FASTEST WAY TO SIMPLIFY FRACTIONS IN ONE STEP!!!
Just divide both the numerator and denominator by the GCF
THE PROBLEM WITH THE METHOD:
If you're not comfortable finding the GCF, you're pretty much sunk with this method! :(
THE BEST REASON TO USE THIS METHOD (other than it's a Calif. STAR Key Standard), it truly is the FASTEST :)
So imagine you have a "GCF Magical Voice" in your head...
The voice tells you the GCF of the numerator and the denominator...
You simply use that GCF to divide both the top and bottom of your fraction and you're done in one step!
THE PRIME FACTORIZATION METHOD:
This is sort of using the GCF "incognito" (in disguise)!
Rewrite the numerator and denominator in prime factorization form.
(Use a Factor Tree or Inverted Division to find the Prime Factorization if necessary).
Then simply cross out each common factor on the top and bottom.
(You're actually using the ID Property of Multiplication because each "crossout" is really a quotient of 1!)
You'll be left with the simplified fraction every time!!!!
If you actually multiplied together all your cross-outs, you'd get the GCF...
so you're using the GCF without even computing it!
THE PROBLEM WITH THIS METHOD:
You may think it's a lot of work
THE BEST REASON TO USE THIS METHOD: Although it takes time, everyone can do a Factor Tree or Inverted Division and create the Prime Factorization...
You'll never get the wrong answer with this one!
THE CROSS OUT METHOD:
You simply think of the first number that comes to your mind that "goz-into" both the numerator and the denominator and keep going until it's simplified.
If it's even, most people start with dividing it in half....and then in half again, etc.
This probably takes the longest, but in practice, most people use this method!
THE PROBLEM WITH THIS METHOD: You may think that a fraction is simplified, but you've missed a factor...this especially happens when the number is odd and you're always used to using 2 to divide the top and the bottom!
THE BEST REASON TO USE THIS METHOD: No one ever forgets how to do this method...it just comes naturally and there are no "precise" steps to do!
EXAMPLE: Simplify by each method: 36/ 54
GCF METHOD:
The GCF is 18:
36 ÷ 18 = 2
54 ÷ 18 = 3
PRIME FACTORIZATION METHOD:
36 = 2 x 2 x 3 x 3
54 = 2 x 3 x 3 x 3
Two of the 3s cross out and one of the 2s
You are now left with:
2/ 3
That's it!!!!!!!!!
CROSS OUT METHOD:
36 ÷ 2 = 18 ÷ 3 = 6 ÷ 3 = 2
54 ÷ 2 = 27 ÷ 3 = 9 ÷ 3 = 3
so 36/54 = 2/3
Do the same thing with variables!
Monday, December 13, 2010
Pre Algebra (Period 2 & 4)
Prime Factorization & GCF 4-3
Greatest Common Factor - think of it backwards to understand it!
Factor = must be a number that goes into the numbers
Common = must be a number that goes into BOTH the numbers
Greatest = must be the biggest number that goes into BOTH the numbers
There are several ways to find it.
1) List all the factors of each number and circle the biggest one that is common to both (takes too long!!)
2) Circle the common factors in the prime factorizations of each number and multiply
3) list the factors in a table and bring down the factors whose column is filled.
Then multiply.
EXAMPLE:
Find the GCF of 36, 45 and 54
LIST ALL THE FACTORS OF EACH NUMBER:
1, 2, 3, 4, 6, 9, 12, 18, 36
1, 3, 5, 9, 15, 45
1, 2, 3, 6, 9, 18, 27, 54
The GCF is 9
FIND THE PRIME FACTORIZATIONS ON A FACTOR TREE OR INVERTED DIVISION AND MULTIPLY THE COMMON FACTORS:
36 = 2 x 2 x 3 x 3
45 = 3 x 3 x 5
54 = 2 x 3 x 3 x 3
GCF = 3 x 3 = 9
PUT THE PRIME FACTORIZATIONS IN A BOX WITH COLUMNS: as shown in class
DO THE SAME THING WITH VARIABLES:
The GCF of the variables is the most of each variable that each term has in common.
EXAMPLE:
Find the GCF of a2b3c4 ac3d a3c2f
The COMMON variables are a and c
How many of each variable is COMMON to all 3 terms:
They each have 1 a (although the first term has 2 and the 3rd term has 3)
They each have 2 c's (although the 1st term has 4 and the 2nd has 3)
GCF = ac2
Again, the GCF of variables is simply the lowest power of common variables
You should look for a special case of GCFs:
When one number goes into the other number(s), the smaller number is always the GCF.
Example: The GCF of 50 and 100 is 50
50 is the biggest factor that goes into both 50 and 100!
Greatest Common Factor - think of it backwards to understand it!
Factor = must be a number that goes into the numbers
Common = must be a number that goes into BOTH the numbers
Greatest = must be the biggest number that goes into BOTH the numbers
There are several ways to find it.
1) List all the factors of each number and circle the biggest one that is common to both (takes too long!!)
2) Circle the common factors in the prime factorizations of each number and multiply
3) list the factors in a table and bring down the factors whose column is filled.
Then multiply.
EXAMPLE:
Find the GCF of 36, 45 and 54
LIST ALL THE FACTORS OF EACH NUMBER:
1, 2, 3, 4, 6, 9, 12, 18, 36
1, 3, 5, 9, 15, 45
1, 2, 3, 6, 9, 18, 27, 54
The GCF is 9
FIND THE PRIME FACTORIZATIONS ON A FACTOR TREE OR INVERTED DIVISION AND MULTIPLY THE COMMON FACTORS:
36 = 2 x 2 x 3 x 3
45 = 3 x 3 x 5
54 = 2 x 3 x 3 x 3
GCF = 3 x 3 = 9
PUT THE PRIME FACTORIZATIONS IN A BOX WITH COLUMNS: as shown in class
DO THE SAME THING WITH VARIABLES:
The GCF of the variables is the most of each variable that each term has in common.
EXAMPLE:
Find the GCF of a2b3c4 ac3d a3c2f
The COMMON variables are a and c
How many of each variable is COMMON to all 3 terms:
They each have 1 a (although the first term has 2 and the 3rd term has 3)
They each have 2 c's (although the 1st term has 4 and the 2nd has 3)
GCF = ac2
Again, the GCF of variables is simply the lowest power of common variables
You should look for a special case of GCFs:
When one number goes into the other number(s), the smaller number is always the GCF.
Example: The GCF of 50 and 100 is 50
50 is the biggest factor that goes into both 50 and 100!
Thursday, December 9, 2010
Algebra (Period 1)
Factoring x2 +bx + c or Factoring Trinomials 6-4
You are reversing it back to BEFORE it was FOILed.
Always check your factoring by FOILing or BOXing back!!
Factoring Trinomials with a:
PLUS sign as the second sign
x2 + bx + c
Following these steps:
1. set up your hugs ( )( )
2. When the last sign is positive the BOTH signs in each of the ( )( ) are the SAME!!
3. How do you know what those two signs are? It is whatever the sign is of the 2nd term of the trinomial. Put that sign in BOTH parentheses.
4. to factor (unFOIL), you will need to find two factors that
MULTIPLY to the LAST term and
ADD to the MIDDLE term
you can set up a box with
___ X ___ =
___ + ___ =
and fill in the blanks.
I suggest you make a T-chart with all the factors of the last term-- using your divisibility rules!!
Example:
x2 +8x + 15
Follow the steps
( )( )
Think: last term sign is + so both signs are the same
Think: first sign (sign of the 2nd term) is + so both signs are positive
put + into the ( )( )
( + )( + )
You already know the "F" in FOIL means that both first terms must be x --> so put those terms in
(x + )(x + )
Now to get to the L in FOIL you need two factors whose product is 15. This is easy but using a T chart
15
1 I 15
3 I 5
you see 1 X 15 or 3 X 5 are possibilities
BUT, you also need two numbers to add to the I and O of FOIL which means that the two numbers must add up to 8 ( the middle term)
Since 3 + 5 = 8, they must be the two factors that will work
3 X 5 = 15
3 + 5 = 8
(x + 5) (x +3)
At this point it does not matter which factor you put into the first ( ) because they are the SAME sign but I always tend to put the LARGER number in the first ( ) because of other rules -- which you will learn later this week)
Next example
x2 - 8x + 15
Follow the steps
( ) ( )
THINK: Last sign is + so the signs are the same
THINK: First sign ( 2nd term) is NEGATIVE so BOTH signs are NEGATIVE
( - ) ( - )
Again,
You already know the "F" in FOIL means that both first terms must be x --> so put those terms in
(x - )(x - )
Now to get the "L" in FOIL, you need two factors whose product is 15
BUT, you also need two numbers to add to the I and O of FOIL which means that the two numbers must add up to -8 ( the middle term)
Since 3 + 5 = 8, they must be the two factors that will work
-3 X -5 = 15
-3 + -5 = -8
(x - 5)(x - 3)
(It doesn't matter which is first because they're the same sign!) Now FOIL to see if we're right!
Last example:
x2 - 8xy + 15y2
Same problem as the one above except now there are two variables. Simply use the same steps above and include the y
(x -5y)(x -3y)
Factoring Trinomials with a:
NEGATIVE sign as the second sign
x2 + bx - c
We will use the same method as yesterday to factor these basic trinomials! 1.
Set up your ( )( )
2. Look at the SECOND or last sign
If it's negative, then the signs in the ( ) are DIFFERENT
Why?
Because when you multiply integers and get a NEGATIVE product, the only way that will happen is if they are DIFFERENT signs.
Remember that the last term is the product of the two LAST terms in FOILing.
3. Now look at the sign of the second term.
It tells you "Who wins," meaning which sign must have the larger absolute value. Remember that the middle term is the SUM of the "O" and the "I" terms when FOILing.
Because these two terms have DIFFERENT signs, when you add them, you actually "subtract" and take the sign of the larger absolute value.
( This is just integer rules!!)
Put that sign in the first parentheses and always put the bigger number in the first parentheses.
4. To UNFOIL (factor), you will need to find 2 FACTORS that MULTIPLY to the last term, but SUBTRACT to the middle term.
(yesterday the factors needed to ADD to the middle)
Or you can still say you're adding, but since they are DIFFERENT signs, you will end up subtracting!
This is still an educated guess and check!
To help you do this, I suggest to set it up like this:
____ x ____ = ____
____ - ____ = ____
Again, setting up a T-chart with all the factors also helps you visualize the two numbers you are looking for!!
EXAMPLE:
x2 + 2x - 15
( ) ( )
THINK: LAST sign is - so the signs are DIFFERENT
THINK: First sign is + so the POSITIVE WINS!!
( + ) ( - )
You know the "F" in FOIL means that both the fist terms must be x so
(x + )(x - )
Now to get the "L" in FOIL, you need 2 factors whose product is NEGATIVE 15
(Don't forget the sign1!!)
Several possibilities like 1 X -15 or 15 X -1 or 3 X -5 or 5 X -3
BUT since the POSITIVE must win , according to the middle term of the example (+2x)
you know that the bigger factor must be positive ( so it can win!!)
Therefore your choices are POSITIVE 15 X NEGATIVE 1 or POSITIVE 5 X NEGATIVE 3
BUT, you also need them to ADD to the I and O in FOIL so pick the two factors that also ADD to POSITIVE 2
Since -3 + 5 = +2 these must be the two factors that will work
I set it up like this:
____ x ____ = 15
____ - ____ = 2
so 5 x 3 = 15
5 - 3 = 2
YOU CAN ALSO DO THIS WITH THE APPROPRIATE SIGNS and adding:
+____ x -____ = -15
+____ + -____ = + 2
so
5 x (-3) = -15
5 +( -3) = 2
THIS IS WHERE IT DOES MATTER WHICH NUMBER YOU DO HAVE WITH THE SIGN BECASUE THE + MUST WIN!!
( x + 5 )( x - 3 )
NEXT EXAMPLE:
x2 - 2x -15
( )( )
THINK: Last sign is - so signs are DIFFERENT!
THINK: First sign is - so NEGATIVE MUST WIN
( - )( + )
You know the the "F" in FOIL means that both first terms must be x
( x - )( x + )
Now to get the "L" in FOIL, you need 2 factors whose product is NEGATIVE 15
Like 1 and 15, or 3 and 5
But you also need to add to the I and O in FOIL which means that the two factors must add to NEGATIVE 2
Since 3 + -5 = -2, this must be the two factors that will work:
( x - 5 )( x + 3)
It matters which number you have with which sign because the negatives must win!
That's why I always put the sign of the middle term in the first parentheses.
That way, I always know to put the larger number in the first parentheses, so that sign will win.
Now FOIL to see if we're right!
LAST EXAMPLE:
x2 - 2xy -15y2
Same problem as the one before, except now there are 2 variables!
Simply use the same factorization and include the y
( x - 5y )( x + 3y)
ALWAYS CHECK BY FOILing or BOXing Back!!
You are reversing it back to BEFORE it was FOILed.
Always check your factoring by FOILing or BOXing back!!
Factoring Trinomials with a:
PLUS sign as the second sign
x2 + bx + c
Following these steps:
1. set up your hugs ( )( )
2. When the last sign is positive the BOTH signs in each of the ( )( ) are the SAME!!
3. How do you know what those two signs are? It is whatever the sign is of the 2nd term of the trinomial. Put that sign in BOTH parentheses.
4. to factor (unFOIL), you will need to find two factors that
MULTIPLY to the LAST term and
ADD to the MIDDLE term
you can set up a box with
___ X ___ =
___ + ___ =
and fill in the blanks.
I suggest you make a T-chart with all the factors of the last term-- using your divisibility rules!!
Example:
x2 +8x + 15
Follow the steps
( )( )
Think: last term sign is + so both signs are the same
Think: first sign (sign of the 2nd term) is + so both signs are positive
put + into the ( )( )
( + )( + )
You already know the "F" in FOIL means that both first terms must be x --> so put those terms in
(x + )(x + )
Now to get to the L in FOIL you need two factors whose product is 15. This is easy but using a T chart
15
1 I 15
3 I 5
you see 1 X 15 or 3 X 5 are possibilities
BUT, you also need two numbers to add to the I and O of FOIL which means that the two numbers must add up to 8 ( the middle term)
Since 3 + 5 = 8, they must be the two factors that will work
3 X 5 = 15
3 + 5 = 8
(x + 5) (x +3)
At this point it does not matter which factor you put into the first ( ) because they are the SAME sign but I always tend to put the LARGER number in the first ( ) because of other rules -- which you will learn later this week)
Next example
x2 - 8x + 15
Follow the steps
( ) ( )
THINK: Last sign is + so the signs are the same
THINK: First sign ( 2nd term) is NEGATIVE so BOTH signs are NEGATIVE
( - ) ( - )
Again,
You already know the "F" in FOIL means that both first terms must be x --> so put those terms in
(x - )(x - )
Now to get the "L" in FOIL, you need two factors whose product is 15
BUT, you also need two numbers to add to the I and O of FOIL which means that the two numbers must add up to -8 ( the middle term)
Since 3 + 5 = 8, they must be the two factors that will work
-3 X -5 = 15
-3 + -5 = -8
(x - 5)(x - 3)
(It doesn't matter which is first because they're the same sign!) Now FOIL to see if we're right!
Last example:
x2 - 8xy + 15y2
Same problem as the one above except now there are two variables. Simply use the same steps above and include the y
(x -5y)(x -3y)
Factoring Trinomials with a:
NEGATIVE sign as the second sign
x2 + bx - c
We will use the same method as yesterday to factor these basic trinomials! 1.
Set up your ( )( )
2. Look at the SECOND or last sign
If it's negative, then the signs in the ( ) are DIFFERENT
Why?
Because when you multiply integers and get a NEGATIVE product, the only way that will happen is if they are DIFFERENT signs.
Remember that the last term is the product of the two LAST terms in FOILing.
3. Now look at the sign of the second term.
It tells you "Who wins," meaning which sign must have the larger absolute value. Remember that the middle term is the SUM of the "O" and the "I" terms when FOILing.
Because these two terms have DIFFERENT signs, when you add them, you actually "subtract" and take the sign of the larger absolute value.
( This is just integer rules!!)
Put that sign in the first parentheses and always put the bigger number in the first parentheses.
4. To UNFOIL (factor), you will need to find 2 FACTORS that MULTIPLY to the last term, but SUBTRACT to the middle term.
(yesterday the factors needed to ADD to the middle)
Or you can still say you're adding, but since they are DIFFERENT signs, you will end up subtracting!
This is still an educated guess and check!
To help you do this, I suggest to set it up like this:
____ x ____ = ____
____ - ____ = ____
Again, setting up a T-chart with all the factors also helps you visualize the two numbers you are looking for!!
EXAMPLE:
x2 + 2x - 15
( ) ( )
THINK: LAST sign is - so the signs are DIFFERENT
THINK: First sign is + so the POSITIVE WINS!!
( + ) ( - )
You know the "F" in FOIL means that both the fist terms must be x so
(x + )(x - )
Now to get the "L" in FOIL, you need 2 factors whose product is NEGATIVE 15
(Don't forget the sign1!!)
Several possibilities like 1 X -15 or 15 X -1 or 3 X -5 or 5 X -3
BUT since the POSITIVE must win , according to the middle term of the example (+2x)
you know that the bigger factor must be positive ( so it can win!!)
Therefore your choices are POSITIVE 15 X NEGATIVE 1 or POSITIVE 5 X NEGATIVE 3
BUT, you also need them to ADD to the I and O in FOIL so pick the two factors that also ADD to POSITIVE 2
Since -3 + 5 = +2 these must be the two factors that will work
I set it up like this:
____ x ____ = 15
____ - ____ = 2
so 5 x 3 = 15
5 - 3 = 2
YOU CAN ALSO DO THIS WITH THE APPROPRIATE SIGNS and adding:
+____ x -____ = -15
+____ + -____ = + 2
so
5 x (-3) = -15
5 +( -3) = 2
THIS IS WHERE IT DOES MATTER WHICH NUMBER YOU DO HAVE WITH THE SIGN BECASUE THE + MUST WIN!!
( x + 5 )( x - 3 )
NEXT EXAMPLE:
x2 - 2x -15
( )( )
THINK: Last sign is - so signs are DIFFERENT!
THINK: First sign is - so NEGATIVE MUST WIN
( - )( + )
You know the the "F" in FOIL means that both first terms must be x
( x - )( x + )
Now to get the "L" in FOIL, you need 2 factors whose product is NEGATIVE 15
Like 1 and 15, or 3 and 5
But you also need to add to the I and O in FOIL which means that the two factors must add to NEGATIVE 2
Since 3 + -5 = -2, this must be the two factors that will work:
( x - 5 )( x + 3)
It matters which number you have with which sign because the negatives must win!
That's why I always put the sign of the middle term in the first parentheses.
That way, I always know to put the larger number in the first parentheses, so that sign will win.
Now FOIL to see if we're right!
LAST EXAMPLE:
x2 - 2xy -15y2
Same problem as the one before, except now there are 2 variables!
Simply use the same factorization and include the y
( x - 5y )( x + 3y)
ALWAYS CHECK BY FOILing or BOXing Back!!
Pre Algebra (Period 2 & 4)
Prime Factorization & GCF 4-3
Prime Factorization Now that you know what a number divides by,
you can get its prime factorization
Prime = a number with exactly 2 factors (itself and 1)
Composite = a number with at least 3 factors
1 and 0 are neither composite nor prime
How do you find all the prime factors of a number?
You learned Factor Trees in previous years and Factor Trees are a good strategy
Make sure you take all the bottom factors only!
EXAMPLE: 54
The prime factorization of 54 = 2x3x3x3 or 2(33)
Make sure you list the factors from least to greatest!
There are multiple Factor Trees based on what you decide to use as your first 2 factors. But each one will get you to the same "bottom" of the tree.
The prime factorization of 54 IS ALWAYS = 2x3x3x3 or 2(33) no matter how you start your Factor Tree
You can also use something called INVERTED DIVISION.
This is similar to Factor Trees but you must begin with the SMALLEST PRIME FACTOR that goes into the number and keep using it until it no longer works.
Then you go on to the next higher prime factor, etc.
EXAMPLE: For 54 again, but this time using Inverted Division.
THERE IS ONLY ONE CORRECT FORM OF INVERTED DIVISION!
BUT... the factors will always be in order if you follow the rules!!
Prime Factorization Now that you know what a number divides by,
you can get its prime factorization
Prime = a number with exactly 2 factors (itself and 1)
Composite = a number with at least 3 factors
1 and 0 are neither composite nor prime
How do you find all the prime factors of a number?
You learned Factor Trees in previous years and Factor Trees are a good strategy
Make sure you take all the bottom factors only!
EXAMPLE: 54
The prime factorization of 54 = 2x3x3x3 or 2(33)
Make sure you list the factors from least to greatest!
There are multiple Factor Trees based on what you decide to use as your first 2 factors. But each one will get you to the same "bottom" of the tree.
The prime factorization of 54 IS ALWAYS = 2x3x3x3 or 2(33) no matter how you start your Factor Tree
You can also use something called INVERTED DIVISION.
This is similar to Factor Trees but you must begin with the SMALLEST PRIME FACTOR that goes into the number and keep using it until it no longer works.
Then you go on to the next higher prime factor, etc.
EXAMPLE: For 54 again, but this time using Inverted Division.
THERE IS ONLY ONE CORRECT FORM OF INVERTED DIVISION!
BUT... the factors will always be in order if you follow the rules!!
Wednesday, December 8, 2010
Pre Algebra (Period 2 & 4)
Exponents 4-2
26 represents 2⋅2⋅2⋅2⋅2⋅2 = 64
the 2 in 26 is the base
the 6 in 26 is the exponent
and together 26 it is the power!!
The base is used as a factor 6 times to produce 64
There are 3 forms here:
26 is in exponential notation
2⋅2⋅2⋅2⋅2⋅2 is in expanded notation
64 is in standard notation
You must use integer rules-- even when raising a negative number to a power!!
odd power = negative
even power = positive
(-5)3 = (-5)(-5)(-5) = -125
(-5)4 = (-5)(-5)(-5)(-5) = 625
an odd number of negative signs or an odd power---> it's negative
an even number of negative signs or an even power ---> it's positive
(-2)3 = ( -2)(-2)(-2) = -8
(-2)4 = (-2)(-2)(-2)(-2) = 16
If there is a negative BUT NO parenthesis it is ALWAYS negative
-25
is read as the opposite of 25
Look at what the exponent is touching.. and in this case it is just the 2!!
You can also think of -25 as (-1)-25
so this is -1⋅2⋅2⋅2⋅2⋅2 = -32
-24 = -2⋅2⋅2⋅2 or -1⋅2⋅2⋅2⋅2 = -16
BUT REMEMBER
(-2)4 = (-2)(-2)(-2)(-2) = +16
THis works with variables as well. WHen you substitute in for variables put the number in parenthesis!! Hugs are important in life-- and equally important in math!!
For example Solve for x
x4 - 10 when x = -2
(-2)4 - 10
+16 - 10 = 6
But what happens when the expression is
-x4 - 10? Still solve for x , when x = -2
This time
the exponent is touching just the x
-(-2)4 - 10
You must use PEMDAS and do the exponent first!!
(-2)4 = 16 so substitute that back in
-(16) - 10
becomes
-16 -10 = -26
What about 4(2y -3)2 when y = 5
substitute in
4[2(5) -3]2
4(10-3)2
4(7)2
4(49) =196
How about
-2x3 + 4y . When x = -2 and y = 3
-2(-2)3 + 4(3)
-2(-8) + 12
16 + 12 = 28
26 represents 2⋅2⋅2⋅2⋅2⋅2 = 64
the 2 in 26 is the base
the 6 in 26 is the exponent
and together 26 it is the power!!
The base is used as a factor 6 times to produce 64
There are 3 forms here:
26 is in exponential notation
2⋅2⋅2⋅2⋅2⋅2 is in expanded notation
64 is in standard notation
You must use integer rules-- even when raising a negative number to a power!!
odd power = negative
even power = positive
(-5)3 = (-5)(-5)(-5) = -125
(-5)4 = (-5)(-5)(-5)(-5) = 625
an odd number of negative signs or an odd power---> it's negative
an even number of negative signs or an even power ---> it's positive
(-2)3 = ( -2)(-2)(-2) = -8
(-2)4 = (-2)(-2)(-2)(-2) = 16
If there is a negative BUT NO parenthesis it is ALWAYS negative
-25
is read as the opposite of 25
Look at what the exponent is touching.. and in this case it is just the 2!!
You can also think of -25 as (-1)-25
so this is -1⋅2⋅2⋅2⋅2⋅2 = -32
-24 = -2⋅2⋅2⋅2 or -1⋅2⋅2⋅2⋅2 = -16
BUT REMEMBER
(-2)4 = (-2)(-2)(-2)(-2) = +16
THis works with variables as well. WHen you substitute in for variables put the number in parenthesis!! Hugs are important in life-- and equally important in math!!
For example Solve for x
x4 - 10 when x = -2
(-2)4 - 10
+16 - 10 = 6
But what happens when the expression is
-x4 - 10? Still solve for x , when x = -2
This time
the exponent is touching just the x
-(-2)4 - 10
You must use PEMDAS and do the exponent first!!
(-2)4 = 16 so substitute that back in
-(16) - 10
becomes
-16 -10 = -26
What about 4(2y -3)2 when y = 5
substitute in
4[2(5) -3]2
4(10-3)2
4(7)2
4(49) =196
How about
-2x3 + 4y . When x = -2 and y = 3
-2(-2)3 + 4(3)
-2(-8) + 12
16 + 12 = 28
Algebra (Period 1)
Trinomial Squares 6-3
This is a special product that we learned in Chapter 5 when we did FOILing.
FOIL: (a + 3)2 (called a binomial squared)
(a + 3)(a + 3) = a2 + 3a + 3a + 9
a2 + 6a + 9 (called a trinomial square)
Again, you see that the middle term is DOUBLE the product of the two terms in the binomial, and the first and last terms are simply the squares of each term in the binomial.
HOW TO RECOGNIZE THAT IT IS A BINOMIAL SQUARED:
1) Is it a trinomial? (if it's a binomial, it cannot be a binomial squared - it may be diff of 2 squares)
2) Are the first and last terms POSITIVE?
3) Are the first and last terms perfect squares?
4) Is the middle term double the product of the square roots of the first and last terms?
IF YES TO ALL OF THESE QUESTIONS, THEN YOU HAVE A TRINOMIAL SQUARE
TO FACTOR A TRINOMIAL SQUARE: a2 + 6a + 9
1) ( )2
2) Put the sign of the middle term in the ( + )2
3) Find the square root of the first term and the last term and place in the parentheses:
(3 + a)2
4) Check by FOILing back. (or using the BOX method)
x2 - 14x + 49
(x -7)2
16x2 - 56xy + 49y2
ask yourself those important questions. They are all YES... so
(4x - 7y)2
x2 -4xy + 4y2
(x - 2y)2
y6 + 16y3 + 64
(y3 + 8)2
9a8 - 30a4b + 25b2
(3a4 -5b)2
What about
2x2 -40x + 200
What's the first thing you must look for? ALWAYS!!! the GCF
2(x2 -20x + 100) now it is a trinomial square
2(x -10)2
similarly with
2x2 -4x + 2
2(x2 -2x + 1)... again NOW it is a trinomial square
2(x -1)2
18x3 + 12x2 + 2x
What's the GCF? 2x
2x(9x2 + 6x + 1)
2x(3x + 1)2
But look at
(a +4)2 - 2(a +4) + 1
How can that be a trinomial square?
Well, think about the following
A2 + 2AB + B2 and
A2 - 2AB + B2
represent generic trinomial squares
so if we let (a + 4) represent A... it works
[(a +4) -1]2
but we can simplify that to
(a +3)2
You could multiply everything out and then combine like terms,
that is,
(a +4)2 - 2(a +4) + 1 becomes
a2 -8a + 16 -2a -8 + 1 which then simplifies
a2 -6a + 9 which is definitely a trinomial square
(a-3)2
but that's what we got using a simpler method!!
Try
(y+3)2 + 2(y+3) + 1
It is a trinomial square in the form
A2 + 2AB + B2
so [(y + 3) +1]2
which becomes (y +4)2
This is a special product that we learned in Chapter 5 when we did FOILing.
FOIL: (a + 3)2 (called a binomial squared)
(a + 3)(a + 3) = a2 + 3a + 3a + 9
a2 + 6a + 9 (called a trinomial square)
Again, you see that the middle term is DOUBLE the product of the two terms in the binomial, and the first and last terms are simply the squares of each term in the binomial.
HOW TO RECOGNIZE THAT IT IS A BINOMIAL SQUARED:
1) Is it a trinomial? (if it's a binomial, it cannot be a binomial squared - it may be diff of 2 squares)
2) Are the first and last terms POSITIVE?
3) Are the first and last terms perfect squares?
4) Is the middle term double the product of the square roots of the first and last terms?
IF YES TO ALL OF THESE QUESTIONS, THEN YOU HAVE A TRINOMIAL SQUARE
TO FACTOR A TRINOMIAL SQUARE: a2 + 6a + 9
1) ( )2
2) Put the sign of the middle term in the ( + )2
3) Find the square root of the first term and the last term and place in the parentheses:
(3 + a)2
4) Check by FOILing back. (or using the BOX method)
x2 - 14x + 49
(x -7)2
16x2 - 56xy + 49y2
ask yourself those important questions. They are all YES... so
(4x - 7y)2
x2 -4xy + 4y2
(x - 2y)2
y6 + 16y3 + 64
(y3 + 8)2
9a8 - 30a4b + 25b2
(3a4 -5b)2
What about
2x2 -40x + 200
What's the first thing you must look for? ALWAYS!!! the GCF
2(x2 -20x + 100) now it is a trinomial square
2(x -10)2
similarly with
2x2 -4x + 2
2(x2 -2x + 1)... again NOW it is a trinomial square
2(x -1)2
18x3 + 12x2 + 2x
What's the GCF? 2x
2x(9x2 + 6x + 1)
2x(3x + 1)2
But look at
(a +4)2 - 2(a +4) + 1
How can that be a trinomial square?
Well, think about the following
A2 + 2AB + B2 and
A2 - 2AB + B2
represent generic trinomial squares
so if we let (a + 4) represent A... it works
[(a +4) -1]2
but we can simplify that to
(a +3)2
You could multiply everything out and then combine like terms,
that is,
(a +4)2 - 2(a +4) + 1 becomes
a2 -8a + 16 -2a -8 + 1 which then simplifies
a2 -6a + 9 which is definitely a trinomial square
(a-3)2
but that's what we got using a simpler method!!
Try
(y+3)2 + 2(y+3) + 1
It is a trinomial square in the form
A2 + 2AB + B2
so [(y + 3) +1]2
which becomes (y +4)2
Tuesday, December 7, 2010
Math 6 Honors (Period 6 and 7)
Triangles 4-4
A triangle is the figure formed when three points, not on a line are jointed by segments.
Triangle ABC ΔABC
Each of the Points A, B, C is called a vertex
(plural: Vertices) of ΔABC
Each of the angles angle A. angle B. and angle C is called an angle of ΔABC
In any triangle-
The sum of the lengths of any two sides is greater than the length of the third side
The sum of the measures of the angles is 180
There are several ways to name triangles. One way is by angles
Acute Triangle
3 acute angles
Right Triangle
1 right angle
Obtuse Triangle
1 obtuse angle
Triangles can be classified by their sides
Scalene Triangle
no 2 sides congruent
Isosceles Triangle
at least 2 sides congruent
Equilateral Triangle
all 3 sides congruent
The longest side of a triangle is opposite the largest angle and the shortest side is opposite the smallest angle. Two angles are congruent if and only if the sides opposite them are congruent.
A triangle is the figure formed when three points, not on a line are jointed by segments.
Triangle ABC ΔABC
Each of the Points A, B, C is called a vertex
(plural: Vertices) of ΔABC
Each of the angles angle A. angle B. and angle C is called an angle of ΔABC
In any triangle-
The sum of the lengths of any two sides is greater than the length of the third side
The sum of the measures of the angles is 180
There are several ways to name triangles. One way is by angles
Acute Triangle
3 acute angles
Right Triangle
1 right angle
Obtuse Triangle
1 obtuse angle
Triangles can be classified by their sides
Scalene Triangle
no 2 sides congruent
Isosceles Triangle
at least 2 sides congruent
Equilateral Triangle
all 3 sides congruent
The longest side of a triangle is opposite the largest angle and the shortest side is opposite the smallest angle. Two angles are congruent if and only if the sides opposite them are congruent.
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