# 2.2.2: Graphs of Polynomials Using Zeros

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

## Graphs of Polynomials Using Zeros

How is finding and using the zeroes of a higher-degree polynomial related to the same process you have used in the past on quadratic functions?

## Graphing Polynomials Using Zeros

The following procedure can be followed when graphing a polynomial function.

• Use the leading-term test to determine the end behavior of the graph.
• Find the x−intercept(s) of f(x) by setting f(x)=0 and then solving for x.
• Find the y−intercept of f(x) by setting y=f(0) and finding y.
• Use the x−intercept(s) to divide the x−axis into intervals and then choose test points to determine the sign of f(x) on each interval.
• Plot the test points.
• If necessary, find additional points to determine the general shape of the graph.

## The Leading-Term Test

If anxn is the leading term of a polynomial. Then the behavior of the graph as x→∞ or x→−∞ can be known by one the four following behaviors:

2. If an<0 and n even: 3. If an>0 and n odd: 4. If an<0 and n odd: ## Examples

###### Example 1

Earlier, you were asked to identify some similarities in graphing using zeroes between quadratic functions and higher-degree polynomials.

Solution

Despite the more complex nature of the graphs of higher-degree polynomials, the general process of graphing using zeroes is actually very similar. In both cases, your goal is to locate the points where the graph crosses the x or y axis. In both cases, this is done by setting the y value equal to zero and solving for x to find the x axis intercepts, and setting the x value equal to zero and solving for y to find the y axis intercepts.

###### Example 2

Find the roots (zeroes) of the polynomial:

h(x)=x3+2x2−5x−6

Solution

Start by factoring:

h(x)=x3+2x2−5x−6=(x+1)(x−2)(x+3)

To find the zeros, set h(x)=0 and solve for x.

(x+1)(x−2)(x+3)=0

This gives

x+1=0

x−2=0

x+3=0

or

x=-1

x=2

x=-3

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

###### Example 3

Find the zeros of g(x)=−(x−2)(x−2)(x+1)(x+5)(x+5)(x+5).

Solution

The polynomial can be written as

g(x)=−(x−2)2(x+1)(x+5)3

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

−(x−2)2(x+1)(x+5)3=0

this gives

x−2=0

x+1=0

x+5=0

or

x=2

x=-1

x=-5

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

###### Example 4

Graph the polynomial function f(x)=−3x4+2x3.

Solution

Since the leading term here is −3x4 then an=−3<0, and n=4 even. Thus the end behavior of the graph as x→∞ and x→−∞ is that of Box #2, item 2.

We can find the zeros of the function by simply setting f(x)=0 and then solving for x.

−3x4+2x3=0

−x3(3x−2)=0

This gives

x=0 or x=$$\ 2\over 3$$

So we have two x−intercepts, at x=0 and at x=$$\ 2\over 3$$, with multiplicity k=3 for x=0 and multiplicity k=1 for x=$$\ 2\over 3$$

To find the y−intercept, we find f(0), which gives

f(0)=0

So the graph passes the y−axis at y=0.

Since the x−intercepts are 0 and $$\ 2\over 3$$, they divide the x−axis into three intervals: (−∞, 0), (0, $$\ 2\over 3$$), and ($$\ 2\over 3$$, ∞). Now we are interested in determining at which intervals the function f(x) is negative and at which intervals it is positive. To do so, we construct a table and choose a test value for x from each interval and find the corresponding f(x) at that value.

Interval Test Value x f(x) Sign of f(x) Location of points on the graph
(−∞, 0) -1 -5 - below the x−axis
(0, $$\ 2\over 3$$) $$\ 1\over 2$$ $$\ 1\over 16$$ + above the x−axis
($$\ 2\over 3$$, ∞) 1 -1 - below the x−axis

Those test points give us three additional points to plot: (−1, −5), ($$\ 1\over 2$$, $$\ 1\over 16$$), 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 x→−∞ and x→+∞. 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.

###### Example 5

Find the zeros and sketch a graph of the polynomial

f(x)=x4−x2−56

Solution

This is a factorable equation,

f(x)=x4−x2−56

=(x2−8)(x2+7)

Setting f(x)=0,

(x2−8)(x2+7)=0

the first term gives

x2−8=0

x2=0

x= ±$$\ \sqrt{8}$$

= ±$$\ 2\sqrt{2}$$

and the second term gives

x2+7=0

x2=−7

x=±$$\ \sqrt{-7}$$

=±$$\ i\sqrt{7}$$

So the solutions are ±$$\ 2\sqrt{2}$$ and ±$$\ i\sqrt{7}$$, a total of four zeros of f(x). Keep in mind that only the real zeros of a function correspond to the x−intercept of its graph.

###### Example 6

Graph g(x)=−(x−2)2(x+1)(x+5)3.

Solution

Use the zeros to create a table of intervals and see whether the function is above or below the x−axis in each interval:

Interval Test value x g(x) Sign of g(x) Location of graph relative to x−axis
(−∞, −5) -6 320 + Above
x=−5 -5 0 NA
(-5, -1) -2 144 + Above
x=−1 -1 0 NA
(-1, 2) 0 -100 - Below
x=2 2 0 NA
(2, ∞) 3 -256 - Below

Finally, use this information and the test points to sketch a graph of g(x).

## Review

1. If c is a zero of f, then c is a/an _________________________ of the graph of f.
2. If c is a zero of f, then (x - c) is a factor of ___________________?
3. Find the zeros of the polynomial: P(x)=x3−5x2+6x

Consider the function: f(x)=−3(x−3)4(5x−2)(2x−1)3(4−x)2.

1. How many zeros (x-intercepts) are there?
2. What is the leading term?

Find the zeros and graph the polynomial. Be sure to label the x-intercepts, y-intercept (if possible) and have correct end behavior. You may use technology for questions 9-12.

1. P(x)=−2(x+1)2(x−3)
2. P(x)=x3+3x2−4x−12
3. f(x)=−2x3+6x2+9x+6
4. f(x)=−4x2−7x+3
5. f(x)=2x5+4x3+8x2+6x
6. f(x)=x4−3x2
7. g(x)=x2−|x|
8. Given: P(x)=(3x+2)(x−7)2(9x+2)3

State:

1. The leading term:
2. The degree of the polynomial:
3. The leading coefficient:

Determine the equation of the polynomial based on the graph:

To see the Review answers, open this PDF file and look for section 2.4.

## Vocabulary

Term Definition
Cubic Function A cubic function is a function containing an x3 term as the highest power of x.
Intercept The intercepts of a curve are the locations where the curve intersects the x and y axes. An x intercept is a point at which the curve intersects the x-axis. A y intercept is a point at which the curve intersects the y-axis.
interval An interval is a specific and limited part of a function.
Leading-Term Test The leading-term test is a test to determine the end behavior of a polynomial function.
Polynomial A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.
Polynomial Graph A polynomial graph is the graph of a polynomial function. The term is most commonly used for polynomial functions with a degree of at least three.
Quartic Function A quartic function is a function f(x) containing an x4 term as the highest power of ''x''.
Roots The roots of a function are the values of x that make y equal to zero.
Zeroes The zeroes of a function f(x) are the values of x that cause f(x) to be equal to zero.

This page titled 2.2.2: Graphs of Polynomials Using Zeros is shared under a CK-12 license and was authored, remixed, and/or curated by CK-12 Foundation via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.