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

 Page 3 / 19

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.

#### Questions & Answers

bsc F. y algebra and trigonometry pepper 2
given that x= 3/5 find sin 3x
4
DB
remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
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
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
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
please where is the equation
Miranda
Jeffrey