1. Factoring Polynomials
Objective: To factor polynomials by GCF, special products and grouping terms

2. By GCF Factor each Polynomial
3x4 – 12x3 + 6x2
The GCF of 3x4, -12x3 and 6x2 is 3x2
3x4 – 12x3 + 6x2 = 3x2(x2 – 4x + 2)
8xy2 – 10x2y
The GCF of 8xy2 and -10x2y is 2xy
8xy2 – 10x2y = 2xy(4y – 5x)

3. Perfect Square Trinomials
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2
Example: factor 4x2 – 20x + 25
(2x)2 – 2(2x)(5) + (5)2
(2x – 5)2
Factor x2 + 12x + 36
(x)2 + 2(x)(6) + (6)2
(x + 6)2

4. Difference Of Two Squares
a2 – b2 = (a + b)(a – b)
Example: factor x2 – 49
(x)2 – (7)2 = (x + 7)(x – 7)
Factor: 25h2 – 9k2
(5h)2 – (3k)2 = (5h + 3k)(5h – 3k)

5. Sum And Difference Of Cubes
a3 + b3 = (a + b)(a2 – ab + b2)
a3 – b3 = (a – b)(a2 + ab + b2)
Example: factor x3 – 8
(x)3 – (2)3 = (x – 2)(x2 + (x)(2) + (2)2) = (x – 2)(x2 + 2x + 4)
Factor x3 + 27y3
(x)3 + (3y)3 = (x +3y)(x2 – x(3y) + (3y)2) = (x + 3y)(x2 – 3xy +9y2)

6. Factoring By Grouping
In many instances a polynomial will not be a special product, but may be able to be factored by rearranging its terms.
Factor 2xy – 3 – x + 6y
2xy – x + 6y – 3
x(2y – 1) + 3(2y – 1)
(2y – 1)(x + 3)
Note: this process often requires some trial and error.

7. Try These! Factor each Polynomial
12x3y2 – 6x2y +9xy3
x2 – 8x + 16
x2 – 64y2
8x y3
8x2y + 12x + 3y + 2xy2
3xy(4x2y – 2x + 3y2)
(x – 4)2
(x – 8y)(x + 8y)
(2x + 5y)(4x2 – 10xy + 25y2)
(4x + y)(2xy + 3)

8. Try These! All these problems will require more than one type of factorization
x4 – 16
x6 – y6
2x2y – 8y3
6x2y + 3xy2 – 6xy – 3y2
(x2 + 4)(x + 2)(x – 2) solution
(x + y)(x – y)(x2 + xy + y2)(x2 – xy + y2) solution
2y(x + 2y)(x – 2y) solution
3y(2x + y)(x – 1) solution

9. x4 – 16 (x2)2 – 42 ((x2) + 4)((x2) – 4) (x2 + 4)((x)2 – (2)2)

10. x6 – y6 (x3)2 – (y3)2 (x3 + y3)(x3 – y3)
(x + y)(x2 – xy + y2)(x – y)(x2 + xy +y2)

11. 2x2y – 8y3 2x2y – 23y3 2y(x2 – 22y2) 2y((x)2 – (2y)2)
2y(x + 2y)(x – 2y)

12. 6x2y + 3xy2 – 6xy – 3y2 3xy(2x + y) – 3y(2x + y) (2x + y)(3xy – 3y)
3y(x – 1)(2x + y)