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

College algebra

Page 3 / 19

Try It

Describe in words and symbols the end behavior of $f (x) = - 5 x^{4} .$

As $x$ approaches positive or negative infinity, $f (x)$ decreases without bound: as $x \to \pm \infty, f (x) \to - \infty$ because of the negative coefficient.

Got questions? Get instant answers now!

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 $r$ of the spill depends on the number of weeks $w$ that have passed. This relationship is linear.

r (w) = 24 + 8 w

We can combine this with the formula for the area $A$ of a circle.

A (r) = π r^{2}

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

\begin{matrix} A (w) & = & A (r (w)) \\ = & A (24 + 8 w) \\ = & π {(24 + 8 w)}^{2} \end{matrix}

Multiplying gives the formula.

A (w) = 576 π + 384 π w + 64 π 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.

A General Note

Polynomial functions

Let $n$ be a non-negative integer. A polynomial function is a function that can be written in the form

f (x) = 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 $a_{i}$ is a coefficient and can be any real number other than zero. Each expression $a_{i} x^{i}$ is a term of a polynomial function .

Identifying polynomial functions

Which of the following are polynomial functions?

\begin{matrix} f (x) & = & 2 x^{3} \cdot 3 x + 4 \\ g (x) & = & - x (x^{2} - 4) \\ h (x) & = & 5 \sqrt{x + 2} \end{matrix}

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

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

Got questions? Get instant answers now!

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.

A General Note

Terminology of polynomial functions

We often rearrange polynomials so that the powers are descending.

Diagram to show what the components of the leading term in a function are. The leading coefficient is a_n and the degree of the variable is the exponent in x^n. Both the leading coefficient and highest degree variable make up the leading term. So the function looks like f(x)=a_nx^n +…+a_2x^2+a_1x+a_0.

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

How To

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

Find the highest power of $x$ to determine the degree function.
Identify the term containing the highest power of $x$ to find the leading term.
Identify the coefficient of the leading term.

Questions & Answers

how to create a software using Android phone

Wiseman Reply

how

basra

what is the difference between C and C++.

Yan Reply

what is software

Sami Reply

software is a instructions like programs

Shambhu

what is the difference between C and C++.

Yan

yes, how?

Hayder

what is software engineering

Ahmad

software engineering is a the branch of computer science deals with the design,development, testing and maintenance of software applications.

Hayder

who is best bw software engineering and cyber security

Ahmad

Both software engineering and cybersecurity offer exciting career prospects, but your choice ultimately depends on your interests and skills. If you enjoy problem-solving, programming, and designing software syste

Hayder

what's software processes

Ntege Reply

I haven't started reading yet. by device (hardware) or for improving design Lol? Here. Requirement, Design, Implementation, Verification, Maintenance.

Vernon

I can give you a more valid answer by 5:00 By the way gm.

Vernon

it is all about designing,developing, testing, implementing and maintaining of software systems.

Ehenew

hello assalamualaikum

Sami

My name M Sami I m 2nd year student

Sami

what is the specific IDE for flutter programs?

Mwami Reply

jegudgdtgd my Name my Name is M and I have been talking about iey my papa john's university of washington post I tagged I will be in

Mwaqas Reply

yes

usman

how disign photo

atul Reply

hlo

Navya

Michael

yes

Subhan

Show the necessary steps with description in resource monitoring process (CPU,memory,disk and network)

samuel Reply

What is software engineering

Tafadzwa Reply

Software engineering is a branch of computer science directed to writing programs to develop Softwares that can drive or enable the functionality of some hardwares like phone , automobile and others

kelvin

if any requirement engineer is gathering requirements from client and after getting he/she Analyze them this process is called

Alqa Reply

The following text is encoded in base 64. Ik5ldmVyIHRydXN0IGEgY29tcHV0ZXIgeW91IGNhbid0IHRocm93IG91dCBhIHdpbmRvdyIgLSBTdGV2ZSBXb3puaWFr Decode it, and paste the decoded text here

Julian Reply

what to do you mean

Vincent

hello

ALI

how are you ?

ALI

What is the command to list the contents of a directory in Unix and Unix-like operating systems

George Reply

how can i make my own software free of cost

Faizan Reply

like how

usman

Hayder

The name of the author of our software engineering book is Ian Sommerville.

Doha Reply

what is software

Sampson Reply

the set of intruction given to the computer to perform a task

Noor

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 11

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask

	4 Identifying the degree and leading coefficient of a polynomial By OpenStax Read Online Course
	2 Identifying end behavior of power functions By OpenStax Read Online Course
	2 Physiotherapy Flashcards Set 2 By Rhodes Start Flashcards
©flickr:	Core Java Unit Test-1(CAJ003) By Abishek Devaraj Start Quiz
	7 AP 07 Axial Skeleton Essay By OpenStax Start Flashcards
	15 Dr. Amberg Pharm quiz By Brooke Delaney Start Exam
	2 Human A&P 2- Test 2 By Madison Christian Start Flashcards
©flickr:	Vocabulary Practice Quiz! By Katie Montrose Start Quiz
©flickr: Clive	Sports Psych. Research Methods By Dakota Bocan Start Flashcards
	9 Biology 09 Cell Communication MCQ By OpenStax Start Quiz
	Business fundamentals By OpenStax Read Online Course
	Social Work midterm By Katy Pratt Start Exam