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"




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)

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!

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!

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!!

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!!

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

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

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.

Algebra (Period 1)

Difference of Two Squares 6-2

Again, remember that FACTORING just UNDOES multiplication.

In this case, the multiplication that you'll be UNDOING is FOILING.

FOIL:
(a + b)(a - b)

You will get:
a² - b²
This is the DIFFERENCE (subtraction) of TWO SQUARES.

Now FACTOR:
a² - b²
You undo the FOILING and get:
(a + b)(a - b)


REMEMBER:
You must have two different signs because that's how the MIDDLE TERM disappears!

You will get ADDITIVE INVERSES which will become ZERO



HOW TO RECOGNIZE THE DIFFERENCE OF TWO SQUARES:

1) Is it a binomial?

2) Is it a difference?

3) Are both terms perfect squares?


IF YES TO ALL 3 QUESTIONS, THEN YOU HAVE A DIFFERENCE OF 2 SQUARES!!

HOW TO FACTOR THE DIFFERENCE OF 2 SQUARES:

1) Double hug  (    )(    )

2) Find square root of each term (sq rt sqrt)(sq rt sq rt)

3) Make one sign positive and one sign negative.
              
(sq rt + sqrt)(sq rt -sq rt)


                           
Of course, they get more complicated! 
We can combine pulling out the GCF with this!

ALWAYS LOOK FOR A GCF TO PULL OUT FIRST!!!!!!

EXAMPLE:

27y² - 48y⁴ 

First, look for a GCF that can be pulled out.

The GCF = 3y²

Factor out the GCF (look at Chapter 6-1):
3y²(9 - 16y² )
NOW YOU HAVE A DIFFERENCE OF TWO SQUARES TO FACTOR:


3y²(3 - 4y)(3 + 4y)


CALLED FACTORING COMPLETELY BECAUSE
 YOU CANNOT FACTOR FURTHER!


Always check your factoring by distributing or FOILing back!


THERE IS NO SUCH THING AS THE SUM OF TWO SQUARES!

a² + b² CANNOT BE FACTORED!!!!!


BUT - b² + a²
= + a² -b²
= (a + b)(a - b)
BECAUSE IT'S JUST SWITCHED (COMMUTATIVE)

Algebra (Period 1)

Factoring Polynomials 6-1


Chapter 5 was a very important building block of Algebra but

CHAPTER 6 IS EVEN MORE IMPORTANT FOR HIGH SCHOOL!!!



REMEMBER THIS KEY CONCEPT:
FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL …SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!
(You'll either distribute or FOIL.)


Factoring is a skill that you must understand to be successful in higher level math!!!

We did a simple version of this back in Chapter 1 and you had a RACE on it!


Factoring is simply UNDOING multiplying


Say you multiplied 5 by 10 and got 50

How would you undo it?
DIVIDE by 5!


So FACTORING uses the concept of DIVIDING.

You're actually undoing the DISTRIBUTIVE PROPERTY.

How?
You look for the most of every common factor....the GCF!

Then you pull out the GCF (divide it out of) from each term,
placing the GCF in front of ( )


EXAMPLE:

FIRST, DISTRIBUTE:
2m²n (2n2 + n + 3)
=
4m²n³ + 2m²n² + 6m²n


Now, pretend you don't want the 2m²n to be distributed anymore...

What should you end up with once you UNDO the Distributive Property?

2m²n (2n² + n + 3)


That's exactly what you started with!

So is it that easy?

Well yes... and no...

Yes because that is the answer
…and
…
No because it was only that easy because I gave you how it started!

You won't know how it started in a real problem!



THIS IS AN EXAMPLE OF THE WAY YOU WOULD USUALLY SEE IT.

The question would say:
FACTOR:
 4m²n³ + 2m²n² + 6m²n


Step 1: What does each term have in common (what is the GCF) ?

They each can be divided by 2m²n


Step 2: Put the GCF in front of a set of ( ) and divide each term by the GCF
2m²n ( 4m²n/2m²n + 2m²n²/2m²n + 6m²n/2m²n)


Step 3: SIMPLIFY and you'll get:

2m²n (2n² + n + 3)



Step 4: Check your answer!!!!!

Always check your factoring of the GCF by distributing back!


(incognito, it should be the same thing)

Another check is to make sure you have factored out the entire GCF.

Look inside the parentheses and ask yourself if there is still any factors in common between the terms.
If there is, then you haven't factored out the GREATEST Common Factor.
For example, let's say in the prior example that you only factored out 2mn instead of 2m²n. You would have:
2mn ( 4m²n³/2mn + 2m²n²/2mn + 6m²n/2mn)


= 2mn(2mn²+ mn + 3m)


If you just check by distributing back, the problem will check.

BUT ...
Look inside the ( ) and notice that each term still has a common factor of m!
So this would not be the fully factored form!
So make sure you always look inside the ( )!!!

RELATIVELY PRIME TERMS -

TERMS WITH NO COMMON FACTORS

THAT MEANS THAT THEY CANNOT BE FACTORED
(GCF = 1)

We say they are "not factorable"

Math 6 Honors (Period 6 and 7)

Angles and Angle Measure 4-3
An angle is a figure formed by two rays with the same endpoints. The common endpoint is called the vertex. The rays are called the sides.
We may name an angle by giving its vertex letter if this is the only angle with that vertex, or my listing letters for points on the two sides with the vertex letter in the middle. We use the symbol from the textbook.

To measure segments we use a rule to mark off unit lengths. To measure angles, we use a protractor that is marked off in units of angle measure called degrees.

To use a protractor, place its center point at the vertex of the angle to be measured and one of its zero points on the side.


We often label angels with their measures. When angles have equal measures we can write m angle A = m angle B
We say that angle A and angle B are congruent angles


If two lines intersect so that the angles they form are all congruent, the lines are perpendicular. We use the symbol that looks like an upside down capital T to mean “is perpendicular to.”



Angles formed by perpendicular lines each have measure of 90° . A 90° angle is called a right angle. A small square is often used to indicate a right angle in a diagram

An acute angle is an angle with measure less than 90°. An obtuse angle has measure between 90° and 180°



Two angles are complementary if the sum of the measures is 90°
Two angles are supplementary if the sum of their measures is 180°

Algebra (Period 1)

Multiplying Polynomials 5-11

To multiply two polynomials, multiply each term of one polynomial by every term of the other. THEN ADD the results.

The textbook shows a column approach, please see page 249 for instructions.
In class we used the BOX method... and then combined terms.

When you multiply a trinomial by a binomial or two trinomials, it gets really tricky!


2 ways:

1) box method

2)column method
(double or triple distributive with columns to combine like terms)




If you have a trinomial times a binomial, it's easier to use the Commutative Property

and make it a binomial times a trinomial:

(x² + x - 1) (x - 1)

switch it to

(x - 1) (x² + x - 1)

Remember the following rules
(A + B)(A + B) = (A + B) ² = A² + 2AB + B²

(A - B)(A - B) = (A - B)² = A² -2AB + B²

(A + B)(A - B)= A²- B²
You can use FOIL to multiply two binomials
remember FOIL is First Terms, Outside Terms, Inside Terms, Last Terms

You can always use FOIL-- or the BOX method but knowing these rules will make computation quicker if you know the above rules!!

Math 6 Honors (Period 6 and 7)

Points, Lines, Planes 4-1
We can describe but CANNOT DEFINE point, line or plane in Geometry

We use a single small dot to represent a point and in class we labeled with a P and we called it Point P
A straight line in Geometry is usually just called a line...
Two points DETERMINE exactly ONE LINE

we connected Point P with Point Q and created Line PQ
We placed this type of arrow ↔ over PQ to show a line
↔
PQ

that would represent the line PQ but we found we could write
↔
QP
and mean the SAME line!!

You can name ANY line with ANY TWO points that fall on that line!! USE only TWO points to name a line!!


Three or more points on the same line are called collinear. Notice the word "line" in collinear.
collinear

Points NOT on a same line are called noncollinear.

We have a RAY if we have an endpoint and it extends through other points. We name the rame by Naming the endpoint FIRST
→
PQ is RAY PQ and it begins at P and goes through Q. It is NOT the SAME as

→
QP which is Ray QP, which begins at Q and goes through P

Segments are parts of lines with TWO ENDPOINTS.
⎯
PQ is a segment with endpoints Point P and POint Q

We then looked at the drawing from page 105 and the class named all of the names for the line in the drawing, all of the rays that existed in the figure as well as the segments. We found that there were just 3 collinear points: A, X, B but that we could name 3 sets of non collinear points
X, Y, B and A, X, Y, AND A, B, Y

Three non-collinear points determine a flat surface called a plane.
We name a Plane by using three of its non collinear points!! Plane ABC was out example.

Lines in the same plane that do not intersect are PARALLEL lines. Two segments or rays are parallel if they are parts of parallel lines.
↔
AB is parallel to

↔
CD

may be written
↔ ↔
ABllCD
using two straight lines to indicate parallel

Parallel lines DO NOT intersect.

Intersecting lines intersect in a single point!!



Planes that do not intersect are called parallel planes... we looked around the room and found examples of parts of planes.. noticing which ones were parallel!! (the floor and ceiling were a great example)
Then we drew the box from PAge 106 and identified parallel segments and lines from that box.

Two non parallel lines that do not intersect are called SKEW LINES.