Testing Potential Zeros of a Polynomial Function

What is a Zero of a Polynomial Function?

A zero of a polynomial function f(x) is a number k such that f(k) = 0. The real number zeros are the x-intercepts of the graph of the function.

How Do We Test a Potential Zero of a Polynomial Function?

The remainder theorem gives a quick way to decide if a number k is a zero of a polynomial function defined by f(x), as follows.

1.Use synthetic division to find f(k).

2.If the remainder is 0, then f(k) = 0 and k is a zero of f(x). If the remainder is not 0, then k is not a zero of f(x).  A zero of f(x) is a root, or solution, of the equation f(x) = 0.

Example

Decide whether the given number k is a zero of f(x).

$f\left(x\right)=x^3-4x^2+9x-6;\:k=1$

Solution

Use synthetic division.

Since the remainder is 0, f(1) = 0, and 1 is a zero of the given polynomial function. An x-intercept of the graph of f(x) is 1, so the graph includes the point (1, 0).

On the next page please complete Homework 3.2.