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

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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.

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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 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. Each product   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.
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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.

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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