Name:
Date:
Topic: Factoring Polynomials (Special Cases & By Grouping)
Essential Question: How can you factor special case trinomials and how
does it compare to multiplying special case binomials?
Warm-Up: Factor each expression
 6x2 + 13x + 5

10x2 + 31x – 14
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Home-Learning Review
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Quiz #11
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Factoring Polynomials
Special Cases & By Grouping
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Difference of Squares
Examples:
z2 – 9
16x2 – 81
24g2 – 6
When factoring using a difference of squares,
look for the following three things:
only 2 terms
minus sign between them
both terms must be perfect squares
If all 3 of the above are true, write two
( ), one with a + sign and one with a – sign:
( + ) ( - ).
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Difference of Squares
A “Difference of Squares”
is a binomial (*2 terms only*)
and it factors like this:
2
2
a  b  (a  b)(a  b)
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Difference of Squares
To factor, express each term as a
square of a monomial then apply
2
2
the rule... a  b  (a  b)(a  b)
2
Ex: x 16 
2
2
x 4 
(x  4)(x  4)
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Lets Try Together
Examples:
z2 – 9
16x2 – 81
24g2 – 6
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Try These
1.
a2 – 16
2.
x2 – 25
3.
4y2 – 16
4.
9y2 – 25
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Factoring Four Term Polynomials by Grouping
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Factor by Grouping
When
polynomials contain four terms, it
is sometimes easier to group like terms in
order to factor.
Your goal is to create a common factor.
You can also move terms around in the
polynomial to create a common factor.
Practice makes you better in recognizing
common factors.
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Factor by Grouping
Example 1:
3xy - 21y + 5x – 35
Factor the first two terms:
3xy - 21y = 3y (x – 7)
Factor the last two terms:
+ 5x - 35 = 5 (x – 7)
The green parentheses are the same so it’s
the common factor
Now you have a common factor
(x - 7) (3y + 5)
FACTOR:
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Factor by Grouping
Example 2:
6mx – 4m + 3rx – 2r
Factor the first two terms:
6mx – 4m = 2m (3x - 2)
Factor the last two terms:
+ 3rx – 2r = r (3x - 2)
 The green parentheses are the same so
it’s the common factor
Now you have a common factor
(3x - 2) (2m + r)
FACTOR:
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Factor by Grouping
Example 3:
15x – 3xy + 4y –20
 Factor the first two terms:
15x – 3xy = 3x (5 – y)
 Factor the last two terms:
+ 4y –20 = 4 (y – 5)
 The green parentheses are opposites so change
the sign on the 4
- 4 (-y + 5) or – 4 (5 - y)
 Now you have a common factor
(5 – y) (3x – 4)
 FACTOR:
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Factoring Completely
Now that we’ve learned all the types of factoring,
we need to remember to use them all.
Whenever it says to factor, you must break down
the expression into the smallest possible
factors.
Let’s review all the ways to factor.
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Types of Factoring
1. Look for GCF first.
2. Count the number of terms:
a) 4 terms – factor by grouping
b) 3 terms – apply strategies of
factorization
> x2 + bx + c or ax2 + bx + c
c)
2 terms look for difference of squares
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Solving Equations by Factoring
1. We know that an equation
must be solved for the
unknown.
2. Up to now, we have only
solved equations with a
degree of 1.
2x + 8 = 4x +6
-2x + 8 = 6
-2x = -2
x=1
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Steps to Solve Equations by Factoring
3. If an equation has a degree of 2 or
higher, we cannot solve it until it has
been factored.
4. You must first get “0” on one side of the
= sign before you try any factoring.
5. Once you have “0” on one side, use all
your rules for factoring to make 2 ( ) or
factors.
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Steps to Solve Equations by Factoring
6. Next, set each factor = 0 and solve for the
unknown.
x2 + 12x = 0
Factor GCF
x(x + 12)(x – 3) = 0
(set each factor = 0, & solve)
x=0
x + 12 = 0
x–3=0
x = - 12
x=3
7.
You now have 3 answers, x = 0, x = -12, and x = 3.
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Time to Practice:
Page 515 – 516 (24, 31, 40)
Page 519 – 521 (1, 2, 47 – 50)
Solve for x:
x2 – 10x – 24
2x2 + 13x + 6
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