How Do You Factor a Trinomial Completely?

# How Do You Factor a Trinomial Completely?

A quadratic trinomial is normally factored as the product of two binomials. Using the generic formula abx^2 + cx + d for the trinomial, its factors are (ax + y) and (bx + z) if y times z equals d and az + yb = c.

1. List possible factors for the first term

Consider the trinomial 2x^2 + 9x - 5. The first term, 2x^2, can only be factored as the product of 2x and x. Therefore, the factors of 2x^2 + 9x - 5 are (2x + y) and (x + z) with y and z still to be determined.

2. List possible factors for the third term

Because the third term is negative, one of the factors must be negative, and the other must be positive. The only possible factors that can have a product of -5 are either 1 and -5 or -1 and 5.

3. Choose the factors that produce a sum or difference that's equal to the second term of the trinomial

There are four possible pairs of binomials: (2x + 5) and (x - 1), (2x - 5) and (x + 1), (2x + 1) and (x - 5), or (2x - 1) and (x + 5).The second term of the trinomial must equal the sum of, or difference between, the products of the outer and inner terms. Since 2x times 5 equals 10x and x times -1 equals -1x, the difference between these two products is 9x, so the only possible combination of binomials that can be the factors of 2x^2 + 9x - 5 are (2x - 1) and (x + 5).

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