3.6 Zeros of polynomial functions

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 $x–k,$ the remainder may be found quickly by evaluating the polynomial function at $k,$ that is, $f\left(k\right)$ 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 $d(x)$ where the degree of $d(x)$ is less than or equal to the degree of $f(x),$ there exist unique polynomials $q(x)$ and $r(x)$ such that

If the divisor, $d(x),$ is $x-k,$ this takes the form

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

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

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 tells us

If $k$ is a zero, then the remainder $r$ is $f(k)=0$ and $f(x)=(x-k)q(x)+0$ or $f(x)=(x-k)q(x).$

Notice, written in this form, $x-k$ is a factor of $f(x).$ We can conclude if $k$ is a zero of $f(x),$ then $x-k$ is a factor of $f(x).$