Least Common Multiple 5-6
Also check out December 4, 2009 posting of LCM!! Here is a review of that lesson...
Let’s look at the nonzero multiples of 8 and 12—listed in order
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72…
Multiples of 12: 12, 24, 36, 48, 60, 72, ….
The numbers 24, 48, and 72, ... are called common multiples of 8 and 12. The least of these multiples is 24 and is therefore called the least common multiple.
LCM(8, 12) = 24
To find the LCM of two whole numbers, we can write out lists of multiples of the two numbers.
Or, we can use prime factorization
Lets find LCM(12, 15)
12 = 22∙3
15 = 3∙5
The LCM will be made up of the greatest power of each factor
LCM will be 22∙3∙5 = 60
The book has a third option or method
you can check out, if you’d like
Let’s find LCM (54, 60)
54= 2∙3∙3∙3 = 2∙33
60 = 2∙2∙3∙5 = 22∙3∙5
The greatest power of 2 that occurs in either prime factorization is 22
The greatest power of 3 that occurs in either prime factorization is 33
The greatest power of 5 that occurs in either prime factorization is 5
Therefore, LCM(54,60) is 22∙33∙5 = 540
REMEMBER:
The GCF (greatest common factor) is a factor. The GCF of two numbers will be either the smaller of the two or smaller than both
The LCM (least common multiple) is a multiple. The LCM of the two numbers will be the largest of the two or larger than both.
Friday, December 11, 2009
Algebra Period 4
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!!
Thursday, December 10, 2009
Pre Algebra Period 1
Rational Numbers 4-6
THE NUMBER SYSTEMS
Natural (counting): 1, 2, 3, ....
Whole: 0, 1, 2, 3, ...
Integers: ...-3, -2, -1, 0, 1, 2, 3, ...
Rational: All numbers that can be written as a fraction or RATIO (decimals-terminating and repeating, fractions, mixed numbers, integers)
Irrational: Decimals that never terminate or repeat - like pi and square root of 2
Real numbers: All the number systems together
Complex or Imaginary Numbers: we don't study these until Algebra II, but they are stated as
a + bi where i is equal to the square root of -1
Again, Rational number = any number that can be expressed as the ratio (fraction) of two integers
a/b, where a and b are both integers and b cannot be zero
b cannot be zero because you cannot divide by zero.....IT'S UNDEFINED!
There are positive and negative rational numbers.
THE NUMBER SYSTEMS
Natural (counting): 1, 2, 3, ....
Whole: 0, 1, 2, 3, ...
Integers: ...-3, -2, -1, 0, 1, 2, 3, ...
Rational: All numbers that can be written as a fraction or RATIO (decimals-terminating and repeating, fractions, mixed numbers, integers)
Irrational: Decimals that never terminate or repeat - like pi and square root of 2
Real numbers: All the number systems together
Complex or Imaginary Numbers: we don't study these until Algebra II, but they are stated as
a + bi where i is equal to the square root of -1
Again, Rational number = any number that can be expressed as the ratio (fraction) of two integers
a/b, where a and b are both integers and b cannot be zero
b cannot be zero because you cannot divide by zero.....IT'S UNDEFINED!
There are positive and negative rational numbers.
Algebra Period 4
Trinomial Squares 6-3
This is a special product that we learned in Chapter 5 when we did FOILing.
FOIL: (3 + a)2 (called a binomial squared)
(3 + a)(3 + a) = 9 + 3a + 3a + a2
= 9 + 6a + a2 (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: 9 + 6a + a2
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.
This is a special product that we learned in Chapter 5 when we did FOILing.
FOIL: (3 + a)2 (called a binomial squared)
(3 + a)(3 + a) = 9 + 3a + 3a + a2
= 9 + 6a + a2 (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: 9 + 6a + a2
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.
Wednesday, December 9, 2009
Math 6H ( Periods 3, 6, & 7)
Greatest Common Factor 5-5
Also check out December 1, 2009 posting of GCF. Here is a review of that lesson...
When the factors in the numbers 30 and 42 are listed, the numbers 1, 2, 3, and 6 appear in both lists
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
These numbers are called common factors of 30 and 42. The number 6 is the greatest of these numbers and is therefore called the greatest common factor of the two numbers.
We write
GCF(30,42) = 6
to denote the greatest common factor of 30 and 42
Find GCF(54, 72)
List the factors of each number
54: 1, 2, 3, 6, 9, 18, 27, 54
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
the common factors are 1, 2, 3, 6, 9, and 18
the greatest number in both lists is 18. Therefore,
GCF(54, 72) = 18
Another way to find the GCF of two numbers is to use prime factorization.
Find GCF (54, 72)
First find the prime factorization of 54 and of 75
54 = 2∙3∙3∙3∙3
72 = 2∙2∙2∙3∙3
Find the greatest power of 2 that occurs in both prime factorization. 2
Find the greatest power of 3 that occurs in both prime factorization 32
Therefore GCF(54,72) = 2∙32 = 18
Try GCF(45, 60) using the prime factorization method
The number 1 is a common factor of any two whole numbers.
If 1 is the GCF, then the two numbers are said to be relatively prime.
Show that 15 and 16 are relatively prime
List the factors of each number
Factors of 15 = 1, 3, 5, 15
Factors of 16: 1, 2, 4, 8, 16
Since the GCF(15, 16) = 1, the two numbers are relatively prime
Also check out December 1, 2009 posting of GCF. Here is a review of that lesson...
When the factors in the numbers 30 and 42 are listed, the numbers 1, 2, 3, and 6 appear in both lists
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
These numbers are called common factors of 30 and 42. The number 6 is the greatest of these numbers and is therefore called the greatest common factor of the two numbers.
We write
GCF(30,42) = 6
to denote the greatest common factor of 30 and 42
Find GCF(54, 72)
List the factors of each number
54: 1, 2, 3, 6, 9, 18, 27, 54
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
the common factors are 1, 2, 3, 6, 9, and 18
the greatest number in both lists is 18. Therefore,
GCF(54, 72) = 18
Another way to find the GCF of two numbers is to use prime factorization.
Find GCF (54, 72)
First find the prime factorization of 54 and of 75
54 = 2∙3∙3∙3∙3
72 = 2∙2∙2∙3∙3
Find the greatest power of 2 that occurs in both prime factorization. 2
Find the greatest power of 3 that occurs in both prime factorization 32
Therefore GCF(54,72) = 2∙32 = 18
Try GCF(45, 60) using the prime factorization method
The number 1 is a common factor of any two whole numbers.
If 1 is the GCF, then the two numbers are said to be relatively prime.
Show that 15 and 16 are relatively prime
List the factors of each number
Factors of 15 = 1, 3, 5, 15
Factors of 16: 1, 2, 4, 8, 16
Since the GCF(15, 16) = 1, the two numbers are relatively prime
Pre Algebra Period 1
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!
Algebra Period 4
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)
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)
Tuesday, December 8, 2009
Algebra Period 4
Factoring Polynomials 6-1
Chapter 5 was a very important chapter that you cannot survive without...
CHAPTER 6 IS EVEN MORE IMPORTANT FOR HIGH SCHOOL!!!
As I stated-- these chapters are the Meat & Potatoes of Algebra!!
REMEMBER:
FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!
(You'll either distribute or FOIL...that's what you learned how to do in Chapter 5!)
CHAPTER 6-1: FACTORING THE GCF
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!
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)
4m2 n3 + 2m2 n2 + 6m2 n
Now, pretend you don't want the 2m2n distributed anymore... What should you end up with once you UNDO the Distributive Property?
2m2 n (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 THE WAY YOU WOULD USUALLY SEE IT.
The question would say:
FACTOR: 4m2 n3 + 2m2 n2 + 6m2 n
Step 1: What does each term have in common (what is the GCF) ?
They each can be divided by 2m2 n
Step 2: Put the GCF in front of a set of ( ) and divide each term by the GCF
2m2 n [ (4m2 n3)/2m2 n + (2m2 n2)/2m2 n + (6m2n/2m2 n]
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)
RELATIVELY PRIME TERMS ARE TERMS WITH NO COMMON FACTORS THAT MEANS THAT THEY CANNOT BE FACTORED (GCF = 1) We say they are "not factorable"
Chapter 5 was a very important chapter that you cannot survive without...
CHAPTER 6 IS EVEN MORE IMPORTANT FOR HIGH SCHOOL!!!
As I stated-- these chapters are the Meat & Potatoes of Algebra!!
REMEMBER:
FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!
(You'll either distribute or FOIL...that's what you learned how to do in Chapter 5!)
CHAPTER 6-1: FACTORING THE GCF
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!
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)
4m2 n3 + 2m2 n2 + 6m2 n
Now, pretend you don't want the 2m2n distributed anymore... What should you end up with once you UNDO the Distributive Property?
2m2 n (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 THE WAY YOU WOULD USUALLY SEE IT.
The question would say:
FACTOR: 4m2 n3 + 2m2 n2 + 6m2 n
Step 1: What does each term have in common (what is the GCF) ?
They each can be divided by 2m2 n
Step 2: Put the GCF in front of a set of ( ) and divide each term by the GCF
2m2 n [ (4m2 n3)/2m2 n + (2m2 n2)/2m2 n + (6m2n/2m2 n]
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)
RELATIVELY PRIME TERMS ARE TERMS WITH NO COMMON FACTORS THAT MEANS THAT THEY CANNOT BE FACTORED (GCF = 1) We say they are "not factorable"
Monday, December 7, 2009
Math 6H ( Periods 3, 6, & 7)
Prime Numbers and Composite Numbers 5-4
A prime number is a positive integer greater than 1 with exactly two factors, 1 and the number itself. The numbers 2, 3, 5, 7 are examples of prime numbers
A composite number is a positive integer greater than 1 with more than two factors. The numbers 4, 6, 8, 9, and 10 are examples of composite numbers.
Since 1 has exactly 1 factor, it is neither prime nor composite.
About 230 BCE Erathosthenes, a Greek Mathematician suggested a way to find prime numbers—up to a specific number. The method is called the Sieve of Eratosthenes because it picks out the prime numbers as a strainer, or sieve, picks out solid particles from a liquid.
You may factor a number into prime factors by using either of the following methods
➢ Inverted short division
➢ Factor tree
Both were shown in class.
Could you start the factor tree differently? If so, would you end up with the same answer?
The prime factors of 42 are the same in either factor tree, except for their order.
Every composite number greater than 1 can be written as a product of prime factors in exactly one way, except for the order of the factors.
When we write 42 as 2 ∙ 3 ∙ 7 this product is called the prime factorization of 42
Notice the order in which prime factorization is written.
Let’s try finding the prime factorization of 60
The prime factorization of 60 = 2 ∙ 2∙ 3 ∙ 5 or 22∙ 3∙ 5
A prime number is a positive integer greater than 1 with exactly two factors, 1 and the number itself. The numbers 2, 3, 5, 7 are examples of prime numbers
A composite number is a positive integer greater than 1 with more than two factors. The numbers 4, 6, 8, 9, and 10 are examples of composite numbers.
Since 1 has exactly 1 factor, it is neither prime nor composite.
About 230 BCE Erathosthenes, a Greek Mathematician suggested a way to find prime numbers—up to a specific number. The method is called the Sieve of Eratosthenes because it picks out the prime numbers as a strainer, or sieve, picks out solid particles from a liquid.
You may factor a number into prime factors by using either of the following methods
➢ Inverted short division
➢ Factor tree
Both were shown in class.
Could you start the factor tree differently? If so, would you end up with the same answer?
The prime factors of 42 are the same in either factor tree, except for their order.
Every composite number greater than 1 can be written as a product of prime factors in exactly one way, except for the order of the factors.
When we write 42 as 2 ∙ 3 ∙ 7 this product is called the prime factorization of 42
Notice the order in which prime factorization is written.
Let’s try finding the prime factorization of 60
The prime factorization of 60 = 2 ∙ 2∙ 3 ∙ 5 or 22∙ 3∙ 5
Pre Algebra Period 1
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!
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