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# 2.2: Polynomial Functions

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Understand and apply methods to find the zeros of a factored polynomial
• Use the zeros of polynomial functions to sketch a graph of the function

## Standard Form of Polynomial Function

You have already studied many different kinds of functions, for example linear functions, constant functions, and quadratic functions. All three these functions belong to a larger group of functions called the polynomial functions.

Polynomial Functions

If \begin{align*}P(x)\end{align*} is a polynomial function, then it is given by

\begin{align*}P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_{1}x+a_{0}\end{align*}

where the coefficients \begin{align*}a_{n}, a_{n-1}, \cdots , a_{1}, a_{0}\end{align*} are real numbers and the exponents are positive integers.

We call the first nonzero coefficient \begin{align*}a_{n}\end{align*} the leading coefficient. The term \begin{align*}a_{n}x^{n}\end{align*} is called the leading term. The degree of the polynomial is \begin{align*}n\end{align*}. For example, the quadratic equation \begin{align*}f(x)=-2x^{2}+3x-5\end{align*} has a leading coefficient of -2, a leading term of \begin{align*}-2x^{2}\end{align*} and of degree \begin{align*}n=2\end{align*}. On the other hand, the polynomial \begin{align*}f(x)=1\end{align*} is a polynomial of with a leading coefficient of 1, and the leading term is \begin{align*}1x^{0}=1\end{align*}, so the degree is \begin{align*}n=0\end{align*}.

A very interesting property of polynomials is that they are all continuous, that is, they have no holes, breaks. Informally, we say you can draw the graph of a continuous function without lifting your pencil from the paper. In calculus you will study function continuity in more depth and you will use the continuity of polynomials to simplify the analysis of other, more complicated functions.

In addition to their continuity, the domain of all polynomial functions is the set of all real numbers. That is, the domain is \begin{align*}x\in(-\infty,\infty)\end{align*}.

Sometimes it is helpful to consider a counterexample. There are many functions that we use in mathematics which are not polynomials. For example the floor function, \begin{align*}y=int(x)\end{align*}, is a function that gives the greatest integer less than or equal to \begin{align*}x\end{align*}, so \begin{align*}int(5.67)=5\end{align*}. This function is NOT a polynomial and it is NOT continuous.

## Power Function (even, odd)

The most simple polynomial is called a power function. A power function is a polynomial of the form \begin{align*}f(x)=ax^{n}\end{align*} where \begin{align*}a\end{align*} is a real number and \begin{align*}n\end{align*} is an integer with \begin{align*}n\ge1\end{align*}.

If \begin{align*}n\end{align*} is even, then the power function is also called “even,” and if \begin{align*}n\end{align*} is odd, then the power function is “odd.” The graphs of the first five power functions are shown below.

Notice that each power function has only one \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}intercept, the origin (0, 0).

The end behavior of a function describes the \begin{align*}y-\end{align*}values as \begin{align*}x\end{align*} gets very large (\begin{align*}x\rightarrow\infty\end{align*} in symbols) or as \begin{align*}x\end{align*} gets very small \begin{align*}(x\rightarrow-\infty)\end{align*}.

• For even powers \begin{align*}n\end{align*}, the power function \begin{align*}f(x)=ax^{n}\end{align*} is U-shaped (like a parabola) and as \begin{align*}x\rightarrow\infty, f(x)\rightarrow\infty\end{align*}. Likewise as \begin{align*}x\rightarrow-\infty, f(x)\rightarrow\infty\end{align*}.
• For odd powers \begin{align*}n\end{align*}, the power function goes from the third quadrant to the first quadrant (like the line \begin{align*}y=x\end{align*}). As \begin{align*}x\rightarrow\infty, f(x)\rightarrow\infty,\end{align*} and as \begin{align*}x\rightarrow-\infty, f(x)\rightarrow-\infty\end{align*}.

As with quadratics and polynomials, the leading coefficient \begin{align*}a\end{align*} changes the vertical “stretching” of power functions.

## Graph Polynomial Functions Using Transformations

Just like quadratics, polynomial functions can be graphed using transformations of a known graph. The basic transformations are vertical and horizontal shifts and reflections about the \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}axis.

Given a polynomial \begin{align*}p(x)\end{align*} and constant real numbers \begin{align*}c\end{align*} and \begin{align*}a\end{align*}

• \begin{align*}p(x)+c\end{align*} is a vertical shift of the graph of \begin{align*}p(x)\end{align*} by \begin{align*}c\end{align*} units up (so the function shifts down if \begin{align*}c<0\end{align*}).
• \begin{align*}p(x-c)\end{align*} is a horizontal shift of the graph of \begin{align*}p(x)\end{align*} by \begin{align*}c\end{align*} units to the right. (So the function shifts left if \begin{align*}c<0\end{align*}).
• \begin{align*}-p(x)\end{align*} is a reflection of the graph of \begin{align*}p(x)\end{align*} about the \begin{align*}x-\end{align*}axis.
• \begin{align*}p(-x)\end{align*} is a reflection of the graph of \begin{align*}p(x)\end{align*} about the \begin{align*}y-\end{align*}axis.
• \begin{align*}ap(x)\end{align*} is a vertical stretch by a multiple of \begin{align*}a\end{align*}.
• \begin{align*}p(ax)\end{align*} is a horizontal compression by a multiple of \begin{align*}a\end{align*}.

Example 1

The graph of \begin{align*}f(x)\end{align*} is shown below. Use the graph of \begin{align*}f(x)\end{align*} to graph each of the following: 1) \begin{align*}f(x)+3\end{align*}, 2) \begin{align*}f(x+4)\end{align*}, and 3) \begin{align*}f(-x)+3\end{align*}

Solution

1) This is a vertical shift of \begin{align*}f(x)\end{align*} up by 3 units.

2) This is a horizontal shift of \begin{align*}f(x)\end{align*} to the left by 4 units.

3) This is a reflection of \begin{align*}f(x)\end{align*} about the \begin{align*}y-\end{align*}axis and a vertical shift up by 3 units.

## Graph Polynomial Functions Given Zeros (\begin{align*}x-\end{align*}intercepts)

Recall that the zeros or the roots of a polynomial are the \begin{align*}x-\end{align*}intercepts or the solution set of the polynomial function. For example, the polynomial

\begin{align*}h(x)=x^{3}+2x^{2}-5x-6\end{align*}

can be factored into

\begin{align*}y=h(x)=x^{3}+2x^{2}-5x-6=(x+1)(x-2)(x+3)\end{align*}

To find the zeros, we set

\begin{align*}h(x)=y=0\end{align*}

and solve for \begin{align*}x\end{align*}.

\begin{align*}(x+1)(x-2)(x+3)=0\end{align*}

This gives

\begin{align*}x+1 & = 0\\ x-2 & = 0\\ x+3 & = 0\end{align*}

or

\begin{align*}x & = -1\\ x & = 2\\ x & = -3\end{align*}

So we say that the solution set is \begin{align*}\{-3, -1, 2\}\end{align*}. They are the zeros of the function \begin{align*}h(x)\end{align*}. The zeros of \begin{align*}h(x)\end{align*} are the \begin{align*}x-\end{align*}intercepts of the graph \begin{align*}y=h(x)\end{align*} below.

The previous example illustrates a strategy for graphing a polynomial if you know the zeros of the function. If you know the zeros of a polynomial you can use “test values” between the zeros to see whether the function is above or below the \begin{align*}x-\end{align*}aixs in the interval between the zeros. While this cannot tell you the hieght (\begin{align*}y-\end{align*}values) of the function between the zeros, you can use the zeros to sketch an approximation of the graph. The following examples illustrate first finding the zeros, and then a method for graphing a polynomial once you know the zeros.

Example 2

Find the zeros of \begin{align*}g(x)=-(x-2)(x-2)(x+1)(x+5)(x+5)(x+5)\end{align*}.

Solution

The polynomial can be written as

\begin{align*}g(x)=-(x-2)^{2}(x+1)(x+5)^{3}\end{align*}

To solve the equation, we simply set it equal to zero

\begin{align*}-(x-2)^{2}(x+1)(x+5)^{3}=0\end{align*}

this gives

\begin{align*}x-2 & = 0\\ x+1 & = 0\\ x+5 & = 0\end{align*}

or

\begin{align*}x & = 2\\ x & = -1\\ x & = -5\end{align*}

Notice the occurrence of the zeros in the function. The factor \begin{align*}(x-2)\end{align*} occurred twice (because it was squared), the factor \begin{align*}(x+1)\end{align*} occurred once and the factor \begin{align*}(x+5)\end{align*} occurred three times. We say that the zero we obtain from the factor \begin{align*}(x-2)\end{align*} has a multiplicity \begin{align*}k=2\end{align*} and the factor \begin{align*}(x+5)\end{align*} has a multiplicity \begin{align*}k=3\end{align*}.

To graph \begin{align*}g(x)\end{align*}, use the zeros to create a table of intervals and see whether the function is above or below the \begin{align*}x-\end{align*}axis in each interval:

Interval Test value \begin{align*}x\end{align*} \begin{align*}g(x)\end{align*} Sign of \begin{align*}g(x)\end{align*} Location of graph relative to \begin{align*}x-\end{align*}axis
\begin{align*}(-\infty, -5)\end{align*} -6 320 + Above
\begin{align*}x=-5\end{align*} -5 0 NA
(-5, -1) -2 144 + Above
\begin{align*}x=-1\end{align*} -1 0 NA
(-1, 2) 0 -100 - Below
\begin{align*}x=2\end{align*} 2 NA
\begin{align*}(2, \infty)\end{align*} 3 -256 - Below

Finally, use this information and the test points to sketch a graph of \begin{align*}g(x)\end{align*}.

Example 3

Find the zeros and sketch a graph of the polynomial

\begin{align*}f(x)=x^{4}-x^{2}-56\end{align*}

Solution

This is a factorable equation,

\begin{align*}f(x) & =x^4-x^2-56\\ & = (x^2-8)(x^2+7)\end{align*}

Setting \begin{align*}f(x)=0\end{align*},

\begin{align*}(x^{2}-8)(x^{2}+7) = 0\end{align*}

the first term gives

\begin{align*}x^{2}-8 & = 0\\ x^2 & = 8\\ x & = \pm \sqrt{8}\\ & = \pm 2\sqrt{2}\end{align*}

and the second term gives

\begin{align*}x^{2}+7 & = 0\\ x^2 & = -7\\ x & = \pm \sqrt{-7}\\ & = \pm i\sqrt{7}\end{align*}

So the solutions are \begin{align*}\pm2\sqrt{2}\end{align*} and \begin{align*}\pm i\sqrt{7}\end{align*}, a total of four zeros of \begin{align*}f(x)\end{align*}.

Keep in mind that only the real zeros of a function correspond to the \begin{align*}x-\end{align*}intercept of its graph. For our case in this example, only the two zeros \begin{align*}\pm2\sqrt{2}\end{align*} correspond to actual \begin{align*}x-\end{align*}intercepts (Figure 9) but \begin{align*}+i\sqrt{7}\end{align*} and \begin{align*}-i\sqrt{7}\end{align*} do not, since they are complex. These are given more attention later in the book ((Add cross-referencing link?))

## Summarize Analysis and Graphing of Polynomial Functions

It is helpful to consider the following facts when graphing any polynomial function.

Two Basic Facts About the Graphs of Polynomial Functions

If \begin{align*}f(x)\end{align*} is a polynomial function with degree \begin{align*}n\ge1\end{align*}, then

• The maximum number of real zeros (\begin{align*}x-\end{align*}intercepts) is \begin{align*}n\end{align*}.
• The maximum number of turning points is \begin{align*}n-1\end{align*}.

It is also helpful to learn about the behavior of the function as \begin{align*}x\end{align*} becomes very large \begin{align*}(x\to+\infty)\end{align*} or very small \begin{align*}(x\to-\infty)\end{align*}.

If \begin{align*}a_{n}x^{n}\end{align*} is the leading term of a polynomial. Then the behavior of the graph as \begin{align*}x\to\infty\end{align*} or \begin{align*}x\to-\infty\end{align*} can be known by one the four following behaviors:

1. If \begin{align*}a_{n}>0\end{align*} and \begin{align*}n\end{align*} even

2. If \begin{align*}a_{n}<0\end{align*} and \begin{align*}n\end{align*} even

3. If \begin{align*}a_{n}>0\end{align*} and \begin{align*}n\end{align*} odd

4. If \begin{align*}a_{n}<0\end{align*} and \begin{align*}n\end{align*} odd

The Leading Term Test implies that if you “zoom out” far enough, then all polynomials look like the power function made by the leading term of the polynomial. That is, if the exponent of the leading term is even, then the polynomial is U-shaped, and if the exponent of the leading term is odd, then the polynomial is shaped like \begin{align*}y=x^{3}\end{align*}.

Zeros of a Polynomial Function

• Every polynomial function with degree \begin{align*}n\ge1\end{align*} has at least one zero and at most \begin{align*}n\end{align*} zeros (counting imaginary or complex zeros).

or

• Every polynomial function with degree \begin{align*}n\ge1\end{align*} has exactly \begin{align*}n\end{align*} zeros, if and only if the multiplicities are taken into account (again, counting imaginary zeros).

Example 4

Graph the polynomial function \begin{align*}f(x)=-3x^{4}+2x^{3}\end{align*}.

Solution

Since the leading term here is \begin{align*}-3x^{4}\end{align*} then \begin{align*}a_{n}=-3<0\end{align*}, and \begin{align*}n=4\end{align*} even. Thus the end behavior of the graph as \begin{align*}x\to\infty\end{align*} and \begin{align*}x\to-\infty\end{align*} is that of Box #2, item 2.

We can find the zeros of the function by simply setting \begin{align*}f(x)=0\end{align*} and then solving for \begin{align*}x\end{align*}.

\begin{align*}-3x^{4}+2x^{3} & = 0\\ -x^3(3x-2) & = 0\end{align*}

This gives

\begin{align*}x=0\quad \text{or} \quad x=\frac{2}{3}\end{align*}

So we have two \begin{align*}x-\end{align*}intercepts, at \begin{align*}x=0\end{align*} and at \begin{align*}x=\frac{2}{3}\end{align*}, with multiplicity \begin{align*}k=3\end{align*} for \begin{align*}x=0\end{align*} and multiplicity \begin{align*}k=1\end{align*} for \begin{align*}x=\frac{2}{3}\end{align*}.

To find the \begin{align*}y-\end{align*}intercept, we find \begin{align*}f(0)\end{align*}, which gives

\begin{align*}f(0)=0\end{align*}

So the graph passes the \begin{align*}y-\end{align*}axis at \begin{align*}y=0\end{align*}.

Since the \begin{align*}x-\end{align*}intercepts are 0 and \begin{align*}\frac{2}{3}\end{align*}, they divide the \begin{align*}x-\end{align*}axis into three intervals: \begin{align*}(-\infty, 0), \left ( 0, \frac{2}{3} \right ),\end{align*} and \begin{align*}\left ( \frac{2}{3}, \infty \right )\end{align*}. Now we are interested in determining at which intervals the function \begin{align*}f(x)\end{align*} is negative and at which intervals it is positive. To do so, we construct a table and choose a test value for \begin{align*}x\end{align*} from each interval and find the corresponding \begin{align*}f(x)\end{align*} at that value.

Interval Test Value \begin{align*}x\end{align*} \begin{align*}f(x)\end{align*} Sign of \begin{align*}f(x)\end{align*} Location of points on the graph
\begin{align*}(-\infty, 0)\end{align*} -1 -5 - below the \begin{align*}x-\end{align*}axis
\begin{align*}\left ( 0, \frac{2}{3} \right )\end{align*} \begin{align*}\frac{1}{2}\end{align*} \begin{align*}\frac{1}{16}\end{align*} + above the \begin{align*}x-\end{align*}axis
\begin{align*}\left ( \frac{2}{3}, \infty \right )\end{align*} 1 -1 - below the \begin{align*}x-\end{align*}axis

Those test points give us three additional points to plot: \begin{align*}(-1, -5), \left ( \frac{1}{2},\frac{1}{16} \right )\end{align*}, and (1, -1). Now we are ready to plot our graph. We have a total of three intercept points, in addition to the three test points. We also know how the graph is behaving as \begin{align*}x\to-\infty\end{align*} and \begin{align*}x\to+\infty\end{align*}. This information is usually enough to make a rough sketch of the graph. If we need additional points, we can simply select more points to complete the graph (Figure 10).

To summarize, the following procedure can be followed when graphing a polynomial function.

Graphing a Polynomial Function

1. Use the leading-term test to determine the end behavior of the graph.
2. Find the \begin{align*}x-\end{align*}intercept(s) of \begin{align*}f(x)\end{align*} by setting \begin{align*}f(x)=0\end{align*} and then solving for \begin{align*}x\end{align*}.
3. Find the \begin{align*}y-\end{align*}intercept of \begin{align*}f(x)\end{align*} by setting \begin{align*}y=f(0)\end{align*} and finding \begin{align*}y\end{align*}.
4. Use the \begin{align*}x-\end{align*}intercept(s) to divide the \begin{align*}x-\end{align*}axis into intervals and then choose test points to determine the sign of \begin{align*}f(x)\end{align*} on each interval.
5. Plot the test points.
6. If necessary, find additional points to determine the general shape of the graph.

Example 5

Graph the polynomial function

\begin{align*}g(x)=2x^{3}+3x^{2}-50x-75\end{align*}

Solution

Notice that the leading term is \begin{align*}2x^{3}\end{align*}, where \begin{align*}n=3\end{align*} odd and \begin{align*}a_{n}=2>0\end{align*}. This tells us that the end behavior will take the shape of item 3 in Box 2. Next we find the \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}intercepts. Setting

\begin{align*}g(x) & = 0\\ 2x^3 + 3x^2 -50x - 75 & = 0\end{align*}

We can factor this polynomial by grouping,

\begin{align*}(2x+3)(x^{2})-(2x+3)(25) & = 0\\ (2x+3)(x^2-25) & = 0\\ (x+5)(x-5)(2x+3) & = 0\end{align*}

The zeros are -5, 5, and \begin{align*}\frac{-3}{2}\end{align*}. And they divide the \begin{align*}x-\end{align*}axis into four intervals:

\begin{align*}(-\infty, -5) \qquad \left ( -5, \frac{-3}{2} \right ) \qquad \left ( \frac{-3}{2}, 5 \right ) \qquad (5, \infty)\end{align*}

The \begin{align*}y-\end{align*}intercept is when \begin{align*}y=g(0)\end{align*}. Thus the graph intercepts the \begin{align*}y-\end{align*}axis at \begin{align*}y=-75\end{align*}.

Now we choose test points from each interval and find \begin{align*}g(x)\end{align*}.

Interval Test value \begin{align*}x\end{align*} Value of \begin{align*}g(x)\end{align*} and its sign Location of points on graph
\begin{align*}(-\infty, -5)\end{align*} -6 -99 below the \begin{align*}x-\end{align*}axis
\begin{align*}\left ( -5, \frac{-3}{2} \right )\end{align*} -2 21 above the \begin{align*}x-\end{align*}axis
\begin{align*}\left ( \frac{-3}{2}, 5 \right )\end{align*} 0 -75 below the \begin{align*}x-\end{align*}axis
\begin{align*}(5, \infty)\end{align*} 6 165 above the \begin{align*}x-\end{align*}axis

From the information obtained, we can roughly sketch the graph (below).

## Applications, Technological Tools

You can use the same tools on a graphing calculator to analyze polynomials that you used to analyze quadratics. In particular, you can use the \begin{align*}Y=\end{align*} menu to graph a polynomial, use WINDOW and ZOOM to set up the view, and then use the CALC menu to find MINIMUM, MAXIMUM, and ZEROs of the function.

Two additional tools that are helpful for analyzing graphs with TI-83/84 calculators are the TRACE and TABLE functions.

If you have graphed a function using \begin{align*}Y=\end{align*} and set the WINDOW settings properly to allow you to see the graph, then you can use TRACE to “read” pairs of \begin{align*}(x, y)\end{align*} coordinates on the graph.

((Insert screen shot of graph with the TRACE function turned on))

While this is useful, a drawback of the TRACE function is that when you use the arrow keys ( < and > ) to “trace” along the function values you cannot easily control the amount by which the \begin{align*}x-\end{align*}coordinate changes each time you press the arrow key. One solution to find a particular value of the function is to press TRACE and then enter a number on the keypad. For example the sequence TRACE 5 will show the coordinates of the selected function when \begin{align*}x=5\end{align*}. (Note: this assumes that \begin{align*}x\end{align*} is between XMIN and XMAX. If this is not the case, you will get an error).

((Insert screen shot of function with TRACE 5. It should show \begin{align*}X=5\end{align*} in the lower left corner, and the corresponding \begin{align*}y-\end{align*}value in the lower right))

A third way to analyze a function is to use the table. If you enter 2nd GRAPH then you see a table of function values.

((insert screen shot of table))

If you go to TABLSET (2ND WINDOW), you can set the interval on the table, or change the independent variable to ASK and then type in \begin{align*}x-\end{align*}values directly in the table.

## Exercises

1. Sketch a graph of each power function using the properties of the power functions.
1. \begin{align*}f(x)=-3x^{4}\end{align*}
2. \begin{align*}h(x)=\frac{1}{2}x^{5}\end{align*}
3. \begin{align*}q(x)=4x^{8}\end{align*}
2. Consider the polynomial \begin{align*}f(x)=x^{3}-7x^{2}+10x\end{align*}.
1. Find the zeros (\begin{align*}x-\end{align*}intercepts) of \begin{align*}f(x)\end{align*}.
2. Make a table to show on what intervals \begin{align*}f(x)\end{align*} is positive, and on what intervals \begin{align*}f(x)\end{align*} is negative
3. Sketch a graph of \begin{align*}f(x)\end{align*}
3. Without graphing the function, what is the maximum number of roots and turning points of \begin{align*}g(x)=\frac{2}{3}x^{5}-4x^{3}+7x-\pi\end{align*}. Justify your answer
4. Without graphing the function, what can you say about the end behavior of \begin{align*}g(x)=\frac{2}{3}x^{5}-4x^{3}+7x-\pi\end{align*}? What happens as \begin{align*}x\rightarrow\infty\end{align*}? What happens as \begin{align*}x\rightarrow-\infty\end{align*}?
5. Use a graphing calculator to approximate the zeros of \begin{align*}k(x)=-3x^{4}+18x^{2}-5x+3\end{align*}
6. The graph of \begin{align*}r(x)\end{align*} is shown below Sketch each of the following transformations of \begin{align*}r(x)\end{align*}.
1. \begin{align*}-r(x)\end{align*}
2. \begin{align*}r(x-3)\end{align*}
3. \begin{align*}r(-x)-1\end{align*}
7. Which of the transformations of \begin{align*}r(x)\end{align*} in problem 6 have the same set of zeros as \begin{align*}r(x)\end{align*}?

1. \begin{align*}\{0, 5, 2\}\end{align*}
Interval ( \begin{align*}0 \begin{align*}2 \begin{align*}x>5\end{align*}
Sign of \begin{align*}f(x)\end{align*} - + - +
1. The maximum number of roots of \begin{align*}g(x)\end{align*} is 5 because the degree of \begin{align*}g(x)\end{align*} is 5. The maximum number of turning points is 4.
2. The end behavior of \begin{align*}g(x)\end{align*} is the same as the end behavior of \begin{align*}x^{5}\end{align*}. This is because the leading term of \begin{align*}g(x)\end{align*} is \begin{align*}\frac{2}{3}x^{5}\end{align*}.
3. \begin{align*}\{-2.6052, 2.33885\}\end{align*}
4. Only the transformation in part (a), \begin{align*}-r(x)\end{align*} leaves the zeros the same. The other transformations involve vertical or horizontal shifts which change the \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}intercepts.

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