Factoring Polynomials

If you know your times tables, factoring polynomials can be a straight forward task. Just as you know that the factors of 27 are nine and three, so will you come to recognized the factors of most common polynomials. Here's a simplified approach to the five most common types. With this sheet beside you, identify the type of polynomial you are factoring and follow the appropriate model. Pretty soon you won't have any more problems factoring polynomials.

Difference of Squares
a2 - b2

factors to

(a - b)(a + b).

Note how the middle ab terms cancel one another out. Verify that's the case by multiplying your factored form to see that it works back to the polynomial you started with.

Here's an example.

x2 - 16 = (x - 4)(x + 4)

Factoring the type a2 +2ab + b2
a2 +2ab + b2

factors to

(a + b)(a + b).

Note how the middle ab terms add to one another. Verify that's the case by multiplying your factored form to see that it works back to the polynomial you started with.

Here's an example.

x2 + 8x + 16 = (x + 4)(x + 4)

Factoring the type a2 - 2ab + b2
a2 -2ab + b2

factors to

(a - b)(a - b).

Note that a negative times a negative is a positive. Also note how the middle negative ab terms are collected. Verify that's the case by multiplying your factored form to see that it works back to the polynomial you started with.

Here's an example.

x2 - 8x + 16 = (x - 4)(x - 4)

Factoring the type a3 - b3
a3 - b3

factors to

(a - b)(a2 +ab + b2).

Note that a negative times a positive is a negative. Also note how the middle positive ab term cancels out the middle terms. Verify that's the case by multiplying your factored form to see that it works back to the polynomial you started with.

Here's an example.

x3 - 27 = (x - 3)(x +3x + 9)

Factoring the type a3 + b3
a3 + b3

factors to

(a + b)(a2 -ab + b2).

Note that a negative times a positive is a negative. Also note how the middle positive ab term cancels out the middle terms. Verify that's the case by multiplying your factored form to see that it works back to the polynomial you started with.

Here's an example.

x3 + 27 = (x + 3)(x -3x + 9)