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

Precalculus

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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 $x \to \pm \infty, f (x) \to - \infty$ 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.

\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_{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?

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

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.

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 does the planets on our solar system orbit

cheten Reply

how many Messier objects are there in space

satish Reply

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Richard Reply

what are astronomy

Issan Reply

Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.

Rafael

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what is big bang theory?

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what type of activity astronomer do?

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the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen

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I want to be an astronomer. That's my dream

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Who named the the whole galaxy?

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solar Univers

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what is space

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what is the dark matter

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what are the factors upon which the atmosphere is stratified

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is the big bang the sun

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Sokak

bigbang is the beginning of the universe

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but thats just a theory

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nothing will happen, don't worry brother.

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what does comet means

GANGAIN Reply

these are Rocky substances between mars and jupiter

GANGAIN

Comets are cosmic snowballs of frozen gases , rock and dust that orbit the sun. They are mostly found between the orbits of Venus and Mercury.

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hey can anyone guide me abt international astronomy olympiad

sahil

how can we learn right and true ?

Govinda Reply

why the moon is always appear in an elliptical shape

Gatjuol Reply

Because when astroid hit the Earth then a piece of elliptical shape of the earth was separated which is now called moon.

Hemen

what's see level?

lidiya Reply

Did you mean eye sight or sea level

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oh sorry it's sea level

lidiya

according to the theory of astronomers why the moon is always appear in an elliptical orbit?

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hi !!! I am new in astronomy.... I have so many questions in mind .... all of scientists of the word they just give opinion only. but they never think true or false ... i respect all of them... I believes whole universe depending on true ...থিউরি

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we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe

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another theory of universe except big ban

Albash Reply

how was universe born

Asmit Reply

there many theory to born universe but what is the reality of big bang theory to born universe

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what is the exact value of π?

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by big bang

universal

there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.

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from where on earth could u observe all the stars during the during the course of an year

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I think it couldn't possible on earth

Nagalakshmi

in this time i don't Know

Michele

is that so. the question was in the end of this chapter

Karuna

in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*). So it would just barely creep over the horizon.

Christopher

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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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