Factoring special products

Reading the patterns backwards

Every special product from the polynomials unit, read from right to left, becomes a factoring pattern. The real skill here is recognition: seeing that \(x^2 - 25\) is "a square minus a square", or that \(9y^2 + 6y + 1\) is "the square of a binomial".

That takes an eye for perfect squares and perfect cubes. Keep these handy:

A monomial is a perfect square when its coefficient is a perfect square and every exponent is even: \(36a^6b^2 = (6a^3b)^2\). It's a perfect cube when the coefficient is a perfect cube and every exponent is a multiple of 3: \(8x^9 = (2x^3)^3\).

Difference of squares

Perfect square trinomials

How to spot one: two terms are positive perfect squares, and the remaining term is twice the product of their roots. The sign of that middle term is the sign inside the binomial.

Perfect cubes

Look for four terms: two perfect cubes at the ends, and two middle terms that match the pattern.

Sum and difference of cubes

These two patterns are new. You can verify them by multiplying out the right side.

A memory aid for the signs is SOAP: the binomial has the Same sign as the original, the middle term of the trinomial has the Opposite sign, and the last term is Always Positive.

The trinomial factor looks like a perfect square but has \(ab\) in the middle instead of \(2ab\), so it doesn't factor further.

A factoring strategy

With any polynomial to factor, work through this checklist in order.

Practice