1.5 Factoring polynomials  (Page 3/6)

Factoring a perfect square trinomial

A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.

We can use this equation to factor any perfect square trinomial.

Factoring a difference of squares

A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.

We can use this equation to factor any differences of squares.

Factoring the sum and difference of cubes

Now, we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial.

Similarly, the sum of cubes can be factored into a binomial and a trinomial, but with different signs.

We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of each word relates to the signs: S ame O pposite A lways P ositive. For example, consider the following example.

The sign of the first 2 is the same as the sign between $x^3-2^3.$ The sign of the $2x$ term is opposite the sign between $x^3-2^3.$ And the sign of the last term, 4, is always positive .