# 5.2 Power functions and polynomial functions  (Page 3/19)

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Describe in words and symbols the end behavior of $\text{\hspace{0.17em}}f\left(x\right)=-5{x}^{4}.$

As $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches positive or negative infinity, $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ decreases without bound: as because of the negative coefficient.

## Identifying polynomial functions

An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ of the spill depends on the number of weeks $\text{\hspace{0.17em}}w\text{\hspace{0.17em}}$ that have passed. This relationship is linear.

$r\left(w\right)=24+8w$

We can combine this with the formula for the area $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ of a circle.

$A\left(r\right)=\pi {r}^{2}$

Composing these functions gives a formula for the area in terms of weeks.

$\begin{array}{ccc}\hfill A\left(w\right)& =& A\left(r\left(w\right)\right)\hfill \\ & =& A\left(24+8w\right)\hfill \\ & =& \pi {\left(24+8w\right)}^{2}\hfill \end{array}$

Multiplying gives the formula.

$A\left(w\right)=576\pi +384\pi w+64\pi {w}^{2}$

This formula is an example of a polynomial function . A polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

## Polynomial functions

Let $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ be a non-negative integer. A polynomial function    is a function that can be written in the form

$f\left(x\right)={a}_{n}{x}^{n}+...{a}_{1}x+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$

This is called the general form of a polynomial function. Each $\text{\hspace{0.17em}}{a}_{i}\text{\hspace{0.17em}}$ is a coefficient and can be any real number other than zero. Each expression $\text{\hspace{0.17em}}{a}_{i}{x}^{i}\text{\hspace{0.17em}}$ is a term of a polynomial function    .

## Identifying polynomial functions

Which of the following are polynomial functions?

$\begin{array}{ccc}\hfill f\left(x\right)& =& 2{x}^{3}\cdot 3x+4\hfill \\ \hfill g\left(x\right)& =& -x\left({x}^{2}-4\right)\hfill \\ \hfill h\left(x\right)& =& 5\sqrt{x+2}\hfill \end{array}$

The first two functions are examples of polynomial functions because they can be written in the form $\text{\hspace{0.17em}}f\left(x\right)={a}_{n}{x}^{n}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0},\text{\hspace{0.17em}}$ where the powers are non-negative integers and the coefficients are real numbers.

• $f\left(x\right)\text{\hspace{0.17em}}$ can be written as $\text{\hspace{0.17em}}f\left(x\right)=6{x}^{4}+4.$
• $g\left(x\right)\text{\hspace{0.17em}}$ can be written as $\text{\hspace{0.17em}}g\left(x\right)=-{x}^{3}+4x.$
• $h\left(x\right)\text{\hspace{0.17em}}$ cannot be written in this form and is therefore not a polynomial function.

## Identifying the degree and leading coefficient of a polynomial function

Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power, or in general form. The degree    of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The leading term    is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient    is the coefficient of the leading term.

## Terminology of polynomial functions

We often rearrange polynomials so that the powers are descending.

When a polynomial is written in this way, we say that it is in general form.

Given a polynomial function, identify the degree and leading coefficient.

1. Find the highest power of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ to determine the degree function.
2. Identify the term containing the highest power of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ to find the leading term.
3. Identify the coefficient of the leading term.

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
Navin