# 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.

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
The sequence is {1,-1,1-1.....} has
how can we solve this problem
Sin(A+B) = sinBcosA+cosBsinA
Prove it
Eseka
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
write down the polynomial function with root 1/3,2,-3 with solution
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what is the answer to dividing negative index
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give me the waec 2019 questions