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.

$r(w) \equiv 24+8 w$

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

$A(r)=\pi r^{2}$

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

$\begin{aligned} A(w) &=A(r(w)) \\ &=A(24+8 w) \\ &=\pi(24+8 w)^{2} \end{aligned}$

Multiplying gives the formula.

$A(w)=576 \pi+384 \pi w+64 \pi w^{2}$

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

$f(x)=a_{n} x^{n}+\ldots+a_{2} x^{2}+a_{1} x+a_{0}$

This is called the general form of a polynomial function. Each $a_{i}$ is a coefficient and can be any real number, but $a_{n} \neq$. Each expression $a_{i} x^{i}$ is a term of a polynomial function.

EXAMPLE 4

Identifying Polynomial Functions

Which of the following are polynomial functions?

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

Solution

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

Source: Rice University, https://openstax.org/books/college-algebra/pages/5-2-power-functions-and-polynomial-functions
This work is licensed under a Creative Commons Attribution 4.0 License.

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.

HOW TO

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

EXAMPLE 5

Identifying the Degree and Leading Coefficient of a Polynomial Function

Identify the degree, leading term, and leading coefficient of the following polynomial functions.

$\begin{aligned} f(x) &=3+2 x^{2}-4 x^{3} \\ g(t) &=5 t^{5}-2 t^{3}+7 t \\ h(p) &=6 p-p^{3}-2 \end{aligned}$

Solution

For the function $f(x)$, the highest power of $x$ is $3$, so the degree is $3$. The leading term is the term containing that degree, $-4 x^{3}$. The leading coefficient is the coefficient of that term, $−4$.

For the function $g(t)$, the highest power of $t$ is $5$, so the degree is $5$. The leading term is the term containing that degree, $5 t^{5}$. The leading coefficient is the coefficient of that term, $5$.

For the function $h(p)$, the highest power of $p$ is $3$, so the degree is $3$. The leading term is the term containing that degree, $-p^{3}$. The leading coefficient is the coefficient of that term, $−1$.

TRY IT #3

Identify the degree, leading term, and leading coefficient of the polynomial $f(x)=4 x^{2}-x^{6}+2 x-6$.

Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as $x$ gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the power function consisting of the leading term. See Table 3.

Polynomial Function	Leading Term	Graph of Polynomial Function
$f(x)=5 x^{4}+2 x^{3}-x-4$	$5 x^{4}$
$f(x)=-2 x^{6}-x^{5}+3 x^{4}+x^{3}$	$-2 x^{6}$
$f(x)=3 x^{5}-4 x^{4}+2 x^{2}+1$	$3 x^{5}$
$f(x)=-6 x^{3}+7 x^{2}+3 x+1$	$-6 x^{3}$

Table 3

EXAMPLE 6

Identifying End Behavior and Degree of a Polynomial Function

Describe the end behavior and determine a possible degree of the polynomial function in Figure 7.

Figure 7

Solution

As the input values $x$ get very large, the output values $f(x)$ increase without bound. As the input values $x$ get very small, the output values $f(x)$ decrease without bound. We can describe the end behavior symbolically by writing

$\text { as } x \rightarrow-\infty, f(x) \rightarrow-\infty$

$\text { as } x \rightarrow \infty, f(x) \rightarrow \infty$

In words, we could say that as $x$ values approach infinity, the function values approach infinity, and as $x$ values approach negative infinity, the function values approach negative infinity.

We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.

TRY IT #4

Describe the end behavior, and determine a possible degree of the polynomial function in Figure 8.

Figure 8

EXAMPLE 7

Identifying End Behavior and Degree of a Polynomial Function

Given the function $f(x)=-3 x^{2}(x-1)(x+4)$, express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.

Solution

Obtain the general form by expanding the given expression for $f(x)$.

$\begin{aligned} f(x) &=-3 x^{2}(x-1)(x+4) \\ &=-3 x^{2}\left(x^{2}+3 x-4\right) \\ &=-3 x^{4}-9 x^{3}+12 x^{2} \end{aligned}$

The general form is $f(x)=-3 x^{4}-9 x^{3}+12 x^{2}$. The leading term is $-3 x^{4}$; therefore, the degree of the polynomial is $4$. The degree is even ($4$) and the leading coefficient is negative ($–3$), so the end behavior is

$\text { as } x \rightarrow-\infty, f(x) \rightarrow-\infty$

$\text { as } x \rightarrow \infty, f(x) \rightarrow-\infty$

TRY IT #5

Given the function $f(x)=0.2(x-2)(x+1)(x-5)$, express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.

In addition to the end behavior of polynomial functions, we are also interested in what happens in the "middle" of the function. In particular, we are interested in locations where graph behavior changes. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing.

We are also interested in the intercepts. As with all functions, the $y$-intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one $y$-intercept $\left(0, a_{0}\right)$. The $x$-intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one $x-$intercept. See Figure 9.

Figure 9

INTERCEPTS AND TURNING POINTS OF POLYNOMIAL FUNCTIONS

A turning point of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The $y$-intercept is the point at which the function has an input value of zero. The $x$-intercepts are the points at which the output value is zero.

HOW TO

Given a polynomial function, determine the intercepts.

EXAMPLE 8

Determining the Intercepts of a Polynomial Function

Given the polynomial function $f(x)=(x-2)(x+1)(x-4)$, written in factored form for your convenience, determine the y- and x-intercepts.

Solution

The $y$-intercept occurs when the input is zero so substitute $0$ for $x$.

$\begin{aligned} f(0) &=f(0)=(0-2)(0+1)(0-4) \\ &=(-2)(1)(-4) \\ &=8 \end{aligned}$

The $y$-intercept is $(0, 8)$.

The $x$-intercepts occur when the output is zero.

$0=(x-2)(x+1)(x-4)$

$x-2=0 \quad \text { or } \quad x+1=0 \quad \text { or } \quad x-4=0$

$x=2 \quad \text { or } \quad x=-1 \quad \text { or } \quad x=4$

The $x$-intercepts are $(2,0)$, $(–1,0)$, and $(4,0)$.

We can see these intercepts on the graph of the function shown in Figure 10.

Figure 10

EXAMPLE 9

Determining the Intercepts of a Polynomial Function with Factoring

Given the polynomial function $f(x)=x^{4}-4 x^{2}-45$, determine the $y$- and $x$-intercepts.

Solution

The $y$-intercept occurs when the input is zero.

$\begin{aligned} f(0) &=(0)^{4}-4(0)^{2}-45 \\ &=-45 \end{aligned}$

The $y$-intercept is $(0,−45)$.

The $x$-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.

$\begin{aligned} f(x) &=x^{4}-4 x^{2}-45 \\ &=\left(x^{2}-9\right)\left(x^{2}+5\right) \\ &=(x-3)(x+3)\left(x^{2}+5\right) \end{aligned}$

$0=(x-3)(x+3)\left(x^{2}+5\right)$

$x-3=0 \quad \text { or } \quad x+3=0 \quad \text { or } \quad x^{2}+5=0$

$x=3 \quad \text { or } \quad x=-3 \quad \text { or } \quad \text { (no real solution) }$

The $x$-intercepts are $(3,0)$ and $(–3,0)$.

We can see these intercepts on the graph of the function shown in Figure 11. We can see that the function is even because $f(x)=f(-x)$.

Figure 11

TRY IT #6

Given the polynomial function $f(x)=2 x^{3}-6 x^{2}-20 x$, determine the $y$- and $x$-intercepts.

The degree of a polynomial function helps us to determine the number of $x$-intercepts and the number of turning points. A polynomial function of $n \mathrm{th}$ degree is the product of $n$ factors, so it will have at most $n$ roots or zeros, or $x$-intercepts. The graph of the polynomial function of degree $n$ must have at most $n–1$ turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.

A continuous function has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A smooth curve is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.

INTERCEPTS AND TURNING POINTS OF POLYNOMIALS

A polynomial of degree $n$ will have, at most, $n$ $x$-intercepts and $n−1$ turning points.

EXAMPLE 10

Determining the Number of Intercepts and Turning Points of a Polynomial

Without graphing the function, determine the local behavior of the function by finding the maximum number of x-intercepts and turning points for $f(x)=-3 x^{10}+4 x^{7}-x^{4}+2 x^{3}$.

Solution

The polynomial has a degree of $10$, so there are at most $10$ $x$-intercepts and at most $9$ turning points.

TRY IT #7

Without graphing the function, determine the maximum number of x-intercepts and turning points for $f(x)=108-13 x^{9}-8 x^{4}+14 x^{12}+2 x^{3}$.

EXAMPLE 11

Drawing Conclusions about a Polynomial Function from the Graph

What can we conclude about the polynomial represented by the graph shown in Figure 12 based on its intercepts and turning points?

Figure 12

Solution

The end behavior of the graph tells us this is the graph of an even-degree polynomial. See Figure 13.

Figure 13

The graph has $2$ $x$-intercepts, suggesting a degree of $2$ or greater, and $3$ turning points, suggesting a degree of $4$ or greater. Based on this, it would be reasonable to conclude that the degree is even and at least $4$.

TRY IT #8

What can we conclude about the polynomial represented by the graph shown in Figure 14 based on its intercepts and turning points?

Figure 14

EXAMPLE 12

Drawing Conclusions about a Polynomial Function from the Factors

Given the function $f(x)=-4 x(x+3)(x-4)$, determine the local behavior.

Solution

The $y$-intercept is found by evaluating $f(0)$.

$\begin{aligned} f(0) &=-4(0)(0+3)(0-4)\\ &=0 \end{aligned}$

The $y$-intercept is $(0,0)$.

The $x$-intercepts are found by determining the zeros of the function.

$0=-4 x(x+3)(x-4)$

$x=0 \quad \text { or } \quad x+3=0 \quad \text { or } \quad x-4=0$

$x=0 \quad \text { or } \quad x=-3 \quad \text { or } \quad x=4$

The $x$-intercepts are $(0,0)$, $(–3,0)$, and $(4,0)$.

The degree is $3$ so the graph has at most $2$ turning points.

TRY IT #9

Given the function $f(x) \equiv 0.2(x-2)(x+1)(x-5)$, determine the local behavior.