# 3.6 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 $\text{\hspace{0.17em}}f\left(x\right),\text{\hspace{0.17em}}$ 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

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

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

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 tells us

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

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 $f\left(x\right).\text{\hspace{0.17em}}$

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this