Download Greatest Common Factor and Difference of Squares

Greatest Common
Factor and
Difference of Squares
Greatest Common
Factor
Essential Question
What is the greatest common factor and how does it
relate to quadratic functions?
What is factoring by the GCF?
•The biggest number and highest power of a variable that can
be divided out of all terms.
•Whatever you can take out goes to the front of the
parentheses.
•Your answer should look like a distribution problem.
Example 1
Factor:
2
3x
+ 6x
What is the biggest number that will divide out of each
number?
3
What is the highest power of a variable that can be divided
out of each term?
x1
So, our factored form is: 3x(x + 2)
Example 2
Factor:
3
25y
–
2
30y
What is the biggest number that will divide out of each
number?
5
What is the highest power of a variable that can be divided
out of each term?
y2
So, our factored form is: 5y2(5y – 6)
Example 3
Factor:
3
-6x
–
2
4x
+8
What is the biggest number that will divide out of each
number?
-2 (ALWAYS factor out the negative)
What is the highest power of a variable that can be divided
out of each term?
There isn’t one since 8 doesn’t have a variable.
So, our factored form is: -2(3x3 + 2x2 – 4)
Example 4
Factor:
3
-6x
+
2
15x
+ 11x
What is the biggest number that will divide out of each
number?
-1
What is the highest power of a variable that can be divided
out of each term?
x1
So, our factored form is: -x(6x2 – 15x – 11)
Difference of Squares
Essential Question
What skills that I have already learned will help me
with finding the difference of squares?
What is this method for factoring?
•If there are only 2 terms, check for difference of squares (2 terms that
you can take the square root of).
•Factor like this…
a2 – b2 = (a + b)(a – b)
It will always factor into the sum times the difference of the square
roots.
*Always look for GCF first!
Example 1
Factor:
2
x
–9
•Is there a GCF?
• No.
•Remember: a2 – b2 = (a + b)(a – b)
• a = x and b = 3
•So…factored form is…
• (x + 3)(x – 3)
Example 2
Factor:
2
16x
–
2
4y
•Is there a GCF?
• Yes, 4. So divide both terms: 4(4x2 – y2)
•Factor inside the ( ) using: a2 – b2 = (a + b)(a – b)
• a = 2x and b = y
•So…factored form is…
• 4(2x + y)(2x – y)
Example 3
Factor:
2
25x
–
2
49y
•Is there a GCF?
• No.
•Factor using: a2 – b2 = (a + b)(a – b)
• a = 5x and b = 7y
•So…factored form is…
• (5x + 7y)(5x – 7y)
Example 4
Factor:
2
5x
– 12
•Is there a GCF?
• No.
•5 is not a perfect square so it cannot be factored.
•This is called a “prime polynomial.”
•
Prime
Example 5
Factor:
2
x
+9
•Addition of perfect squares can never be factored!
•
Prime
Example 6
Factor:
4
x
– 16
•Is there a GCF?
• No.
•Remember: a2 – b2 = (a + b)(a – b)
• a = x2 and b = 4
•So…factored form is…
• (x2 + 4)(x2 – 4)
•But the second binomial will factor again…
• (x2 + 4)(x + 2)(x – 2)
Completely factored
Assignment
GCF / Difference of Squares Worksheet
Due Tomorrow!
Reflection
Explain to your partner in your own words what
greatest common factor is.