How to Factor Perfect-Square Trinomials

For a regular trinomial (three-term) expression, you would factor it out by using the FOIL method in reverse. For example, take the expression x^2 - 7x + 12. First look for factors of 12: 1 and 12, 2 and 6, 3 and 4. Pick the pair of factors that add up to -7, in this case -3 and -4. So (x - 3)(x + 4) = x^2 - 7x + 12. For trinomials with perfect squares, here are a few shortcuts you can use.

Instructions

1. Learn the standard forms for perfect square trinomials. (a + b)^2 = a^2 + 2ab + b^2 and (a - b)^2 = a^2 - 2ab + b^2. Memorizing these forms will save crucial amounts of time on standardized tests.

2. A trinomial expression is a perfect square if the first and last terms, properly arranged, are perfect squares and are positive.

3. A trinomial expression is a perfect square if the middle term (regardless of sign) is two times the product of the square roots of the first and third terms.

4. Take an example: x^2 + 8x + 16. This trinomial expression is a perfect square because the first and third terms 1 and 16 are perfect squares. Also, the middle term (8) is twice the product of the square roots of 1 and 16. In other words, 8 is twice 1 and 4.

5. When you factor x^2 + 8x + 16, you get (x + 4)^2. Similarly, as indicated in step 1, x^2 - 8x + 16 = (x - 4)^2.

6. Now for another example: 9x^2 + 6x + 1. This time there is a coefficient (number) in front of x^2. This is a perfect square expression because 9 and 1 are perfect squares. Also, 6 is twice the square roots of 9 and 1. In other words, 6 is twice 3.

7. Factor this perfect square trinomial. 9x^2 + 3x + 1 = (3x + 1)^2. Similarly, 9x^2 - 6x + 1 = (3x - 1)^2.