Algebra and Trigonometry, 5th Edition

4.3 Polynomial Division; The Remainder Theorem and the Factor Theorem

• Perform long division with polynomials and determine whether one polynomial is a factor of another.

• Use synthetic division to divide a polynomial by $x-c$.

• Use the remainder theorem to find a function value $f(c)$.

• Use the factor theorem to determine whether $x-c$ is a factor of $f(x)$.

In general, finding exact zeros of many polynomial functions is neither easy nor straightforward. In this section and the one that follows, we develop concepts that help us find exact zeros of certain polynomial functions with degree 3 or greater.

Consider the polynomial

$$h(x)=x^3+2x^2-5x-6=(x+3)(x+1)(x-2).$$

The factors are

$$x+3,x+1,\text{and}x-2,$$

and the zeros are

$$-3,-1,\text{and}2.$$

When a polynomial is expressed ...