Overview: What Are Square Trinomials?
Square trinomials are polynomials in the form ax2 + bx +c. They have three terms, with the highest degree a squared exponent. They are factored by multiplying binomials.
Perfect Square Trinomials
A perfect square trinomial is the exact square of a trinomial. There are two conditions they follow. The first and last terms must be squares of binomials. Also, the middle term must be twice the product of the monomials. For example, x2 – 14x + 49 and x2 + 14x + 49 are both perfect square trinomials. Both x2 and 49 are perfect squares. In addition, (1 x7) x 2 =14.
Factoring Perfect Square Trinomials
Factoring x 2 -14x + 49 follows those rules. First, find the square root of the outside terms, so that the square root of x2 is x, and the square root of 49 is 7. Because it is a perfect square, the sign between the terms is the same as the sign of the middle term. Therefore, it is (x-7)(x-7). If the trinomial were x2 + 14x +49, it would factor as (x +7)(x +7).
When Trinomials Are Not a Perfect Square
Not all polynomials are a perfect square, even when the first term is squared. These polynomials are very similar. For example, x2 + 5x + 6 can be factored as (x +2)(x +3). Remember the FOIL method? It can be applied backward to factor, as the first term has the factors 1 and x, the outside is 3x, the inside is 2x and the last term, 6, has the factors 2 and 3.
The Difference Between Two Squares Is Also a Square Trinomial
The number sentence a2-b2 can be factored as (a + b)( a- b), as a special type of square trinomial. The middle terms just cancel one another out. Suppose the sentence to expand is 4x2 – 49. It can be factored as (2x -7)(2x +7). The first term is 2x(2x) or 4x2, the outer term is 7(2x) or 14x, the inner term is -7(2x) or -14x, and the last term is -49.
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