Definition of Binomial Factors

By Carlos Mano; Updated April 24, 2017
Binomials are polynomials with two terms.

Polynomials are often the product of smaller polynomial factors. Binomial factors are polynomial factors that have exactly two terms. Binomial factors are interesting because binomials are easy to solve, and the roots of the binomial factors are the same as the roots of the polynomial. Factoring a polynomial is the first step to finding its roots.

Graphing

Graphing a polynomial is a good first step in finding its factors. The points where the graphed curve crosses the X axis are roots of the polynomial. If the curve crosses the axis at point p, then p is a root of the polynomial and X - p is a factor of the polynomial. You should check the factors you get from a graph because it is easy to mistake a reading from a graph. It is also easy to miss multiple roots on a graph.

Candidate Factors

The candidate binomial factors for a polynomial are composed of the combinations of the factors of the first and last numbers in the polynomial. For example 3X^2 - 18X - 15 has as its first number 3, with factors 1 and 3, and as its last number 15, with factors 1, 3, 5 and 15. The candidate factors are X - 1, X + 1, X - 3, X + 3, X - 5, X + 5, X - 15, X + 15, 3X - 1, 3X + 1, 3X - 3, 3X + 3, 3X - 5, 3X + 5, 3X - 15 and 3X + 15.

Finding the Factors

Trying each of the candidate factors, we find that 3X + 3 and X - 5 divide 3X^2 - 18X - 15 with no remainder. So 3X^2 - 18X - 15 = (3X + 3)(X - 5). Notice that 3X + 3 is a factor that we would have missed if we relied on the graph alone. The curve would cross the X axis at -1, suggesting that X - 1 is a factor. Of course, it really is because 3X^2 - 18X - 15 = 3(X + 1)(X - 5).

Finding the Roots

Once you have the binomial factors, it is easy to find the roots of a polynomial -- the roots of the polynomial are the same as the roots of the binomials. For example, the roots of 3X^2 - 18X - 15 = 0 are not obvious, but if you know that 3X^2 - 18X - 15 = (3X + 3)(X - 5), the root of 3X + 3 = 0 is X = -1 and the root of X - 5 = 0 is X = 5.