Factor theorem


In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem.
The factor theorem states that a polynomial has a factor if and only if .

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.
  2. Use the factor theorem to conclude that is a factor of.
  3. Compute the polynomial, for example using polynomial long division or synthetic 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.

    Example

Find the factors of
To do this one would use trial and error to find the first x value that causes the expression to equal zero. To find out if is a factor, substitute into the polynomial above:
As this is equal to 18 and not 0. This means is not a factor of. So, we next try :
This is equal to. Therefore, which is to say, is a factor, and is a root of
The next two roots can be found by algebraically dividing by to get a quadratic:
and therefore and are factors of Of these the quadratic factor can be further factored using the quadratic formula, which gives as roots of the quadratic Thus the three irreducible factors of the original polynomial are and