Three Techniques for Evaluating and Finding Zeros of Polynomial Functions

In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $x–k$, the remainder may be found quickly by evaluating the polynomial function at $k$, that is, $f(k)$. Let's walk through the proof of the theorem.

Recall that the Division Algorithm states that, given a polynomial dividend $f(x)$ and a non-zero polynomial divisor $d(x)$, there exist unique polynomials $q(x)$ and $r(x)$ such that

$f(x)=d(x) q(x)+r(x)$

and either $r(x)=0$ or the degree of $r(x)$ is less than the degree of $d(x)$. In practice divisors, $d(x)$ will have degrees less than or equal to the degree of $f(x)$. If the divisor, $d(x)$, is $x−k$, this takes the form

$f(x)=(x-k) q(x)+r$

Since the divisor $x−k$ is linear, the remainder will be a constant, $r$. And, if we evaluate this for $x=k$, we have

$\begin{aligned} f(k) &=(k-k) q(k)+r \\ &=0 \cdot q(k)+r \\ &=r \end{aligned}$

In other words, $f(k)$ is the remainder obtained by dividing $f(x)$ by $x−k$.

THE REMAINDER THEOREM

If a polynomial $f(x)$ is divided by $x−k$, then the remainder is the value $f(k)$.

HOW TO

Given a polynomial function $f$, evaluate $f(x)$ at $x=k$ using the Remainder Theorem.

1. Use synthetic division to divide the polynomial by $x−k$.

2. The remainder is the value $f(k)$.

EXAMPLE 1

Using the Remainder Theorem to Evaluate a Polynomial

Use the Remainder Theorem to evaluate $f(x)=6 x^{4}-x^{3}-15 x^{2}+2 x-7 \text { at } x=2 .$.

Solution

To find the remainder using the Remainder Theorem, use synthetic division to divide the polynomial by $x−2$.

The remainder is $25$. Therefore, $f(2)=25$.

Analysis

We can check our answer by evaluating $f(2)$.

$\begin{aligned} f(x) &=6 x^{4}-x^{3}-15 x^{2}+2 x-7 \\ f(2) &=6(2)^{4}-(2)^{3}-15(2)^{2}+2(2)-7 \\ &=25 \end{aligned}$

TRY IT #1

Use the Remainder Theorem to evaluate $f(x)=2 x^{5}-3 x^{4}-9 x^{3}+8 x^{2}+2$ at $x=-3$.