Factoring is the process of writing a polynomial as the product of two or more polynomials.

General Strategy for Factoring

• Factor out the greatest common factor
• If the polynomial has two terms, determine whether it matches the pattern of one of the special products:
• Difference of two squares: a - b = (a – b)(a + b)
• Difference of two cubes: a - b = (a – b)(a + ab + b)
• Sum of two cubes: a + b = (a + b)(a - ab + b)
• If the polynomial has three terms, determine whether it is a perfect square trinomial:
• a + 2ab + b = (a + b)
• a - 2ab + b = (a - b)
• If the trinomial is not a perfect square, factor as the product of two binomials using the guess-and-check method or the grouping method.
• If the polynomial has more than three terms, try factoring by grouping.

Example. Factor completely: a)

b)

c)

d) 2y + 10 – 3xy – 15x

e) m - 10m + 16

f) 64 – 27a

g) 12x + 8x – 15

Solution.

a) = 3ac() Factor out greatest

common factor

= 3ac(2a + 3c) Perfect square trinomial

b) = 5rs() Factor out greatest common factor

= 5rs(r – 3s)(r + 3s) Difference of two squares

c) = (3x – 5y) Perfect square trinomial

d) 2y + 10 – 3xy – 15x = 2(y + 5) – 3x(y + 5) Factor by grouping

= (y + 5)(2 – 3x) Factor out common factor, y + 5

e) m - 10m + 16 = (m – 2)(m – 8) Trinomial: guess and check

f) 64 – 27a = () Difference of two cubes

= (4 – 3a)() 4 and 3a substituted into formula

= (4 – 3a)(16 + 12a + 9a) Simplify

g) This example will be done using the grouping method for trinomials. To find

the product number, multiply 12(-15) = -180; the sum number is 8. Two

numbers whose product is –180 and whose sum is 8 are 18 and –10.

12x + 8x – 15 = 12x + 18x – 10x – 15 The 8x term is replaced with

the sum, 18x – 10x, using the numbers found above as coefficients

= 6x(2x + 3) – 5(2x + 3) Factor by grouping

= (2x + 3)(6x – 5) Common factor