1. Home
  2. Algebra and trigonometry
  3. Polynomial and rational functions
  4. Power functions and polynomial

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

<< Chapter < Page
  Algebra and trigonometry     Page 3 / 19
Chapter >> Page >

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 → ± ∞ ,   f ( x ) → − ∞   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.

A ( w ) = A ( r ( w ) ) = A ( 24 + 8 w ) = π ( 24 + 8 w ) 2

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.

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?

f ( x ) = 2 x 3 ⋅ 3 x + 4 g ( x ) = − x ( x 2 − 4 ) h ( x ) = 5 x + 2

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!
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.

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.

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

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

Questions & Answers

calculate molarity of NaOH solution when 25.0ml of NaOH titrated with 27.2ml of 0.2m H2SO4
Gasin Reply
what's Thermochemistry
rhoda Reply
the study of the heat energy which is associated with chemical reactions
Kaddija
How was CH4 and o2 was able to produce (Co2)and (H2o
Edafe Reply
explain please
Victory
First twenty elements with their valences
Martine Reply
what is chemistry
asue Reply
what is atom
asue
what is the best way to define periodic table for jamb
Damilola Reply
what is the change of matter from one state to another
Elijah Reply
what is isolation of organic compounds
IKyernum Reply
what is atomic radius
ThankGod Reply
Read Chapter 6, section 5
Dr
Read Chapter 6, section 5
Kareem
Atomic radius is the radius of the atom and is also called the orbital radius
Kareem
atomic radius is the distance between the nucleus of an atom and its valence shell
Amos
Read Chapter 6, section 5
paulino
Bohr's model of the theory atom
Ayom Reply
is there a question?
Dr
when a gas is compressed why it becomes hot?
ATOMIC
It has no oxygen then
Goldyei
read the chapter on thermochemistry...the sections on "PV" work and the First Law of Thermodynamics should help..
Dr
Which element react with water
Mukthar Reply
Mgo
Ibeh
an increase in the pressure of a gas results in the decrease of its
Valentina Reply
definition of the periodic table
Cosmos Reply
What is the lkenes
Da Reply
what were atoms composed of?
Moses Reply
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!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask