Factoring PolynomialsObjective: To factor polynomials by GCF, special products and grouping terms

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)

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

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)

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)

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.

Try These! Factor each Polynomial

1. 12x3y2 – 6x2y +9xy3

2. x2 – 8x + 16

3. x2 – 64y2

4. 8x3 + 125y3

5. 8x2y + 12x + 3y + 2xy2

• 3xy(4x2y – 2x + 3y2)

• (x – 4)2

• (x – 8y)(x + 8y)

• (2x + 5y)(4x2 – 10xy + 25y2)

• (4x + y)(2xy + 3)

Try These!All these problems will require more than one type of factorization

1. x4 – 16

2. x6 – y6

3. 2x2y – 8y3

4. 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

x4 – 16

• (x2)2 – 42

• ((x2) + 4)((x2) – 4)

• (x2 + 4)((x)2 – (2)2)

• (x2 + 4)(x + 2)(x – 2)

x6 – y6

• (x3)2 – (y3)2

• (x3 + y3)(x3 – y3)

• (x + y)(x2 – xy + y2)(x – y)(x2 + xy +y2)

2x2y – 8y3

• 2x2y – 23y3

• 2y(x2 – 22y2)

• 2y((x)2 – (2y)2)

• 2y(x + 2y)(x – 2y)

6x2y + 3xy2 – 6xy – 3y2

• 3xy(2x + y) – 3y(2x + y)

• (2x + y)(3xy – 3y)

• (3xy – 3y)(2x + y)

• 3y(x – 1)(2x + y)