Factoring Polynomials By Dr. Carol A. Marinas © Copyright 2010 Carol A. Marinas.

Factoring Polynomials

By

Dr. Carol A. Marinas

© Copyright 2010 Carol A. Marinas

Factoring Polynomials

• Greatest Common Factor (GCF)• Difference of Squares• Perfect Square Trinomial• General Trinomials with a = 1 • Perfect Cubes• Four-term Polynomials• General Trinomials with a ≠ 1 • Multiple Factoring Methods


Greatest Common Factor (GCF)

• Remove GCF first

• Example: ax2 – 3a = a(x2 – 3)

• Example 1: 24c – 12d = 12(2c – d)

• Example 2: 3x2 – 6x – 12 = 3(x2 – 2x – 4)


Difference of Perfect Squares

• Subtraction of 2 perfect squares

• Example: a2 – b2 = (a + b) (a – b)

• Example 1: 4c2 – 9 = (2c + 3) (2c – 3)

• Example 2: 49x6y4 – 81d2= (7x3y2 + 9d) (7x3y2 –

9d)


Perfect Square Trinomial

• First and last terms are perfect squares

• Middle term is twice the square root of the product of the first and last term

• Example: a2 + 2ab + b2 =(a + b)2

a2 – 2ab + b2 = (a – b)2

• Example 1: 4d2 – 12d + 9 = (2d – 3)2

• Example 2: 16g4 – 8g2 + 1 = (4g2 – 1)2 = [(2g + 1) (2g – 1)]2 = (2g + 1)2 (2g – 1)2


General Trinomials with a = 1

• In the form: x2 + bx + c

• Look for 2 factors of “c” that also add to “b”

• Example 1: x2 + 5x + 6 = (x + 3) (x + 2)

• Example 2: x2 – x – 42 = (x – 7) (x + 6)


Perfect Cubes

• Sum or Difference of 2 perfect cubes Examples:• a3x3 – b3 = (ax – b) (a2x2 + abx + b2)

• a3x3 + b3 =

(ax + b) (a2x2 – abx + b2) • Same Opposite Always

Plus

• Example 1: 27x3 – 8 = (3x – 2) (9x2 + 6x + 4)

• Example 2: 125b3 + 1 = (5b + 1) (25b2 – 5b +

1)


Four-term Polynomials

• Factor by Grouping

• Example: ax2 + 2ay + 3x2 + 6y

= a(x2 + 2y) + 3(x2 +2y)

= (a + 3) (x2 + 2y)

• Example 1: 7ax2 + 14ag + 5x2 + 10g

= 7a(x2 + 2g) + 5(x2 + 2g)

= (7a + 5) (x2 + 2g)

• Example 2: 6cx2 – 5cx – 12x + 10 =

cx (6x – 5) – 2(6x – 5) = (cx – 2) (6x – 5)


General Trinomial with a ≠ 1

• Example 1: 12c2 – 16c – 3 = 12c2 – 18c + 2c – 3 = 6c(2c – 3) + 1(2c – 3)

= (6c + 1) (2c – 3)

• Example 2: 2x2 + 7x – 15 = 2x2 + 10x – 3x – 15 =

2x(x + 5) – 3(x + 5) = (2x – 3)(x + 5)


Multiple Factoring Methods

• Factor out GCF first

• Then go to other methods

• Example 1: 3x2 – 3y2 = 3(x2 – y2) = 3(x + y) (x – y)

Example 2: 16x3 – 54 = 2(8x3 – 27) = 2(2x – 3) (4x2 + 6x + 9)

Example 3: 2x2 + 12x + 18 = 2(x2 + 6x + 9) = 2(x + 3)2


Factoring Polynomials By Dr. Carol A. Marinas © Copyright 2010 Carol A. Marinas.

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