1. Chapters 8 and 9 Greatest Common Factors & Factoring by Grouping
Definitions
Factor, Factoring, Prime Polynomial
Common Factor of 2 or more terms
Factoring a Monomial into two factors
Identifying Common Monomial Factors
Factoring Out Common Factors
Arranging a 4 Term Polynomial into Groups
Factoring Out Common Binomials

2. What’s a Polynomial Factor?
product = (factor)(factor)(factor) … (factor)
Factoring is the reverse of multiplication.
84 is a product that can be expressed by many different factorizations:
84 = 2(42) or 84 = 7(12) or 84 = 4(7)(3) or 84 = 2(2)(3)(7)
Only one example, 84 = 2(2)(3)(7), shows 84 as the product of prime integers.
Always try to factor a polynomial into prime polynomials

3. Factoring Monomials
12x3 also can be expressed in many ways: 12x3 = 12(x3) 12x3 = 4x2(3x) 12x3 = 2x(6x2)
Usually, we only look for two factors – You try:
4a =
2(2a) or 4(a)
x3 =
x(x2) or x2(x)
14y2 =
14(y2) or 14y(y) or 7(2y2) or 7y(2y) or y(14y)
43x5 =
43(x5) or 43x(x4) or x3(43x2) or 43x2(x3) or …

4. Common Factors of Polynomials
When a polynomial has 2 or more terms, it may have common factors
By definition, a common factor must divide evenly into every term
For x2 + 3x the only common factor is x , so
x2 + 3x = x·x + x·3 = x (? + ?) = x(x + 3)
For 8y2 + 12y – 20 a common factor is 2, so
8y2 + 12y – 20 = 2(? + ? – ?) =2(4y2 + 6y – 10)
Check factoring by multiplying:
2(4y2 + 6y – 10) = 8y2 + 12y – 20

5. The Greatest Common Factor of Polynomials
The greatest common factor (or GCF) is the largest monomial that can divide evenly into every term
Looking for common factors in 2 or more terms … is always the first step in factoring polynomials
Remember a(b + c) = ab + ac (distributive law)
Consider that a is a common factor of ab + ac
If we find a polynomial has form ab + ac we can factor it into a(b + c)
For 3x2 + 3x the greatest common factor is 3x , so
3x2 + 3x = 3x·x + 3x·1 = 3x (? + ?) = 3x(x + 1)
Another example: 8y2 + 12y – 20
The GCF is 4 – Divide each term by 4
8y2 + 12y – 20 = 4(? + ? – ?) = 4(2y2 + 3y – 5)
Check by multiplying: 4(2y2) + 4(3y) – 4(5) = 8y2 + 12y – 20

6. Practice: Find the Greatest Common Monomial Factor
7(? – ?) =
7(a – 3)
19y3 + 3y =
y(? + ?) =
y(19y2 + 3)
8x2 + 14x – 4 =
2(? + ? – ?) =
2(4x2 + 7x – 2)
4y2 + 6y =
2y(? + ?) =
2y(2y + 3)

7. Find the Greatest Common Factor
18y5 – 12y4 + 6y3 =
6y3(? – ? + ?) =
6y3(3y2 – 2y + 1)
21x2 – 42xy + 28y2 =
7(? – ? + ?) =
7(3x2 – 6xy + 4y2)
22x3 – 110xy2 =
22x(? – ?) =
22x(x2 – 5y2)
7x2 – 11xy + 13y2 =
No common factor exists

8. Introduction to Factoring by Grouping: Factoring Out Binomials
x2(x + 7) + 3(x + 7) =
(x + 7)(? + ?) =
(x + 7)(x2 + 3)
y3(a + b) – 2(a + b) =
(a + b)(? – ?) =
(a + b)(y3 – 2)

9. Practice: Factoring Out Binomials
You try: 2x2(x – 1) + 6x(x – 1) – 17(x – 1) =
(x – 1)(? + ? – ?)
(x – 1)(2x2 + 6x – 17)
y2(2y – 5) + x2(2y – 5) =
(2y – 5)(? + ?)
(2y – 5)(y2 + x2)
5x2(xy + 1) + 6y(xy – 1) =
No common factors

10. Factoring by Grouping Example: 2c – 2d + cd – d2 2(c – d) + d(c – d)
For polynomials with 4 terms:
Arrange the terms in the polynomial into 2 groups
Factor out the common monomials from each group
If the binomial factors produced are either identical or opposite, complete the factorization
Example: 2c – 2d + cd – d2
2(c – d) + d(c – d)
(c – d)(2 + d)

11. Factor by Grouping
8t3 + 2t2 – 12t – 3
2t2(4t + 1) – 3(4t + 1)
(4t + 1)(2t2 – 3)

12. Factor by Grouping
4x3 – 6x2 – 6x + 9
2x2(2x – 3) – 3(2x – 3)
(2x – 3)(2x2 – 3)

13. Factor by Grouping y4 – 2y3 – 12y – 3 y3(y – 2) – 3(4y – 1)
Oops – not factorable via grouping

14. Grouping Unusual Polynomials
x3 – 7x2 + 6x + x2y – 7xy + 6y
x(x2 – 7x + 6) + y(x2 – 7x + 6)
(x2 – 7x + 6)(x + y)
(x – 1)(x – 6)(x + y)

15. What Next?
Section 5.6 – Factoring Trinomials