# 5.5 Zeros of polynomial functions  (Page 7/14)

 Page 7 / 14

A shipping container in the shape of a rectangular solid must have a volume of 84 cubic meters. The client tells the manufacturer that, because of the contents, the length of the container must be one meter longer than the width, and the height must be one meter greater than twice the width. What should the dimensions of the container be?

3 meters by 4 meters by 7 meters

Access these online resources for additional instruction and practice with zeros of polynomial functions.

## Key concepts

• To find $\text{\hspace{0.17em}}f\left(k\right),\text{\hspace{0.17em}}$ determine the remainder of the polynomial $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ when it is divided by $\text{\hspace{0.17em}}x-k.\text{\hspace{0.17em}}$ This is known as the Remainder Theorem. See [link] .
• According to the Factor Theorem, $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is a zero of $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ if and only if $\text{\hspace{0.17em}}\left(x-k\right)\text{\hspace{0.17em}}$ is a factor of $\text{\hspace{0.17em}}f\left(x\right).$ See [link] .
• According to the Rational Zero Theorem, each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient. See [link] and [link] .
• When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.
• Synthetic division can be used to find the zeros of a polynomial function. See [link] .
• According to the Fundamental Theorem, every polynomial function has at least one complex zero. See [link] .
• Every polynomial function with degree greater than 0 has at least one complex zero.
• Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form $\text{\hspace{0.17em}}\left(x-c\right),\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ is a complex number. See [link] .
• The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.
• The number of negative real zeros of a polynomial function is either the number of sign changes of $\text{\hspace{0.17em}}f\left(-x\right)\text{\hspace{0.17em}}$ or less than the number of sign changes by an even integer. See [link] .
• Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division. See [link] .

## Verbal

Describe a use for the Remainder Theorem.

The theorem can be used to evaluate a polynomial.

Explain why the Rational Zero Theorem does not guarantee finding zeros of a polynomial function.

What is the difference between rational and real zeros?

Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.

If Descartes’ Rule of Signs reveals a no change of signs or one sign of changes, what specific conclusion can be drawn?

If synthetic division reveals a zero, why should we try that value again as a possible solution?

Polynomial functions can have repeated zeros, so the fact that number is a zero doesn’t preclude it being a zero again.

## Algebraic

For the following exercises, use the Remainder Theorem to find the remainder.

$\left({x}^{4}-9{x}^{2}+14\right)÷\left(x-2\right)$

$\left(3{x}^{3}-2{x}^{2}+x-4\right)÷\left(x+3\right)$

$-106$

$\left({x}^{4}+5{x}^{3}-4x-17\right)÷\left(x+1\right)$

$\left(-3{x}^{2}+6x+24\right)÷\left(x-4\right)$

$\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$

$\left(5{x}^{5}-4{x}^{4}+3{x}^{3}-2{x}^{2}+x-1\right)÷\left(x+6\right)$

what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
Jeffrey
Jeffrey
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function