# 5.5 Zeros of polynomial functions

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In this section, you will:
• Evaluate a polynomial using the Remainder Theorem.
• Use the Factor Theorem to solve a polynomial equation.
• Use the Rational Zero Theorem to find rational zeros.
• Find zeros of a polynomial function.
• Use the Linear Factorization Theorem to find polynomials with given zeros.
• Use Descartes’ Rule of Signs.
• Solve real-world applications of polynomial equations

A new bakery offers decorated sheet cakes for children’s birthday parties and other special occasions. The bakery wants the volume of a small cake to be 351 cubic inches. The cake is in the shape of a rectangular solid. They want the length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of the width. What should the dimensions of the cake pan be?

This problem can be solved by writing a cubic function and solving a cubic equation for the volume of the cake. In this section, we will discuss a variety of tools for writing polynomial functions and solving polynomial equations.

## Evaluating a polynomial using the remainder theorem

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 $\text{\hspace{0.17em}}x–k,\text{\hspace{0.17em}}$ the remainder may be found quickly by evaluating the polynomial function at $\text{\hspace{0.17em}}k,\text{\hspace{0.17em}}$ that is, $\text{\hspace{0.17em}}f\left(k\right)\text{\hspace{0.17em}}$ Let’s walk through the proof of the theorem.

Recall that the Division Algorithm    states that, given a polynomial dividend $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ and a non-zero polynomial divisor $\text{\hspace{0.17em}}d\left(x\right)\text{\hspace{0.17em}}$ where the degree of $\text{\hspace{0.17em}}\text{\hspace{0.17em}}d\left(x\right)\text{\hspace{0.17em}}$ is less than or equal to the degree of $\text{\hspace{0.17em}}f\left(x\right)$ , there exist unique polynomials $\text{\hspace{0.17em}}q\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}r\left(x\right)\text{\hspace{0.17em}}$ such that

$\text{\hspace{0.17em}}f\left(x\right)=d\left(x\right)q\left(x\right)+r\left(x\right)$

If the divisor, $\text{\hspace{0.17em}}d\left(x\right),\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}x-k,\text{\hspace{0.17em}}$ this takes the form

$f\left(x\right)=\left(x-k\right)q\left(x\right)+r$

Since the divisor $\text{\hspace{0.17em}}x-k\text{\hspace{0.17em}}$ is linear, the remainder will be a constant, $\text{\hspace{0.17em}}r.\text{\hspace{0.17em}}$ And, if we evaluate this for $\text{\hspace{0.17em}}x=k,\text{\hspace{0.17em}}$ we have

$\begin{array}{ccc}\hfill f\left(k\right)& =& \left(k-k\right)q\left(k\right)+r\hfill \\ & =& 0\cdot q\left(k\right)+r\hfill \\ & =& r\hfill \end{array}$

In other words, $\text{\hspace{0.17em}}f\left(k\right)\text{\hspace{0.17em}}$ is the remainder obtained by dividing $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ by $\text{\hspace{0.17em}}x-k.\text{\hspace{0.17em}}$

## The remainder theorem

If a polynomial $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is divided by $\text{\hspace{0.17em}}x-k,\text{\hspace{0.17em}}$ then the remainder is the value $\text{\hspace{0.17em}}f\left(k\right).\text{\hspace{0.17em}}$

Given a polynomial function $\text{\hspace{0.17em}}f,$ evaluate $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}x=k\text{\hspace{0.17em}}$ using the Remainder Theorem.

1. Use synthetic division to divide the polynomial by $\text{\hspace{0.17em}}x-k.\text{\hspace{0.17em}}$
2. The remainder is the value $\text{\hspace{0.17em}}f\left(k\right).\text{\hspace{0.17em}}$

## Using the remainder theorem to evaluate a polynomial

Use the Remainder Theorem to evaluate $\text{\hspace{0.17em}}f\left(x\right)=6{x}^{4}-{x}^{3}-15{x}^{2}+2x-7\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}x=2.\text{\hspace{0.17em}}$

To find the remainder using the Remainder Theorem, use synthetic division to divide the polynomial by $\text{\hspace{0.17em}}x-2.\text{\hspace{0.17em}}$

The remainder is 25. Therefore, $\text{\hspace{0.17em}}f\left(2\right)=25.\text{\hspace{0.17em}}$

Use the Remainder Theorem to evaluate $\text{\hspace{0.17em}}f\left(x\right)=2{x}^{5}-3{x}^{4}-9{x}^{3}+8{x}^{2}+2\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}x=-3.\text{\hspace{0.17em}}$

$\text{\hspace{0.17em}}f\left(-3\right)=-412\text{\hspace{0.17em}}$

## Using the factor theorem to solve a polynomial equation

The Factor Theorem is another theorem that helps us analyze polynomial equations. It tells us how the zeros of a polynomial are related to the factors. Recall that the Division Algorithm.

$f\left(x\right)=\left(x-k\right)q\left(x\right)+r$

If $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is a zero, then the remainder $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}f\left(k\right)=0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(x\right)=\left(x-k\right)q\left(x\right)+0\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}f\left(x\right)=\left(x-k\right)q\left(x\right).\text{\hspace{0.17em}}$

Notice, written in this form, $\text{\hspace{0.17em}}x-k\text{\hspace{0.17em}}$ is a factor of $\text{\hspace{0.17em}}f\left(x\right).\text{\hspace{0.17em}}$ We can conclude if $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is a zero of $\text{\hspace{0.17em}}f\left(x\right),\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}x-k\text{\hspace{0.17em}}$ is a factor of $f\left(x\right).\text{\hspace{0.17em}}$

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