Factor Theorem - Factorization of Polynomials

Factorization of Polynomials

Two problems where the factor theorem is commonly applied are those of factoring a polynomial and finding the roots of a polynomial equation; it is a direct consequence of the theorem that these problems are essentially equivalent.

The factor theorem is also used to remove known zeros from a polynomial while leaving all unknown zeros intact, thus producing a lower degree polynomial whose zeros may be easier to find. Abstractly, the method is as follows:

1. "Guess" a zero of the polynomial . (In general, this can be very hard, but math textbook problems that involve solving a polynomial equation are often designed so that some roots are easy to discover.)
2. Use the factor theorem to conclude that is a factor of .
3. Compute the polynomial, for example using polynomial long division.
4. Conclude that any root of is a root of . Since the polynomial degree of is one less than that of, it is "simpler" to find the remaining zeros by studying .