In the previous example, writing a degree polynomial with zeros of [latex]-3,2[/latex] resulted in two answers: [latex]f(x)=(x+3)^{2}(x-2)[/latex] or [latex]f(x)=(x+3)(x-2)^{2}[/latex]. The exponents on each of the factors are called multiplicities.

Multiplicities are useful because they indicate whether a graph touches or crosses the x-axis.

Identify zeros and their multiplicities

Graphs behave differently at various x-intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and bounce off.

Suppose, for example, we graph the function

Notice in Figure 5 that the behavior of the function at each of the x-intercepts is different.

Figure 5. Identifying the behavior of the graph at an x-intercept by examining the multiplicity of the zero.

The x-intercept [latex]x=-3[/latex] is the solution of equation [latex]x+3=0[/latex]. The graph passes directly through the x-intercept at [latex]x=-3[/latex]. The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line—it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.

The x-intercept [latex]x=2[/latex] is the repeated solution of the equation [latex]{\left(x - 2\right)}^{2}=0[/latex]. The graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic—it bounces off of the horizontal axis at the intercept.

The factor is repeated, that is, the factor [latex]\left(x - 2\right)[/latex] appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity. The zero associated with this factor, [latex]x=2[/latex], has multiplicity 2 because the factor [latex]\left(x - 2\right)[/latex] occurs twice.

The x-intercept [latex]x=-1[/latex] is the repeated solution of factor [latex]{\left(x+1\right)}^{3}=0[/latex]. The graph passes through the axis at the intercept, but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic—with the same S-shape near the intercept as the toolkit function [latex]f\left(x\right)={x}^{3}[/latex]. We call this a triple zero, or a zero with multiplicity 3.

For zeros with even multiplicities, the graphs touch or are tangent to the x-axis. For zeros with odd multiplicities, the graphs cross or intersect the x-axis. See Figure 6 for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.

Figure 6

For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the horizontal axis but, for each increasing even power, the graph will appear flatter as it approaches and leaves the x-axis.

For higher odd powers, such as 5, 7, and 9, the graph will still cross through the horizontal axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the x-axis.

Example 7: Identifying Zeros and Their Multiplicities

Use the graph of the function of degree 6 to identify the zeros of the function and their possible multiplicities.

Figure 7

Show Solution

The polynomial function is of degree n. The sum of the multiplicities must be n.

Starting from the left, the first zero occurs at [latex]x=-3[/latex]. The graph touches the x-axis, so the multiplicity of the zero must be even. The zero of –3 has multiplicity 2.

The next zero occurs at [latex]x=-1[/latex]. The graph looks almost linear at this point. This is a single zero of multiplicity 1.

The last zero occurs at [latex]x=4[/latex]. The graph crosses the x-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is likely 3 and that the sum of the multiplicities is likely 6.

Try It

Use the graph of the function of degree 5 to identify the zeros of the function and their multiplicities.

Figure 8

Show Solution

The graph has a zero of –5 with multiplicity 1, a zero of –1 with multiplicity 2, and a zero of 3 with even multiplicity.

 Use factoring to find zeros of polynomial functions

Find zeros of polynomial functions

Recall that if f is a polynomial function, the values of x for which [latex]f\left(x\right)=0[/latex] are called zeros of f. If the equation of the polynomial function can be factored, we can set each factor equal to zero and solve for the zeros.

We can use this method to find x-intercepts because at the x-intercepts we find the input values when the output value is zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to three cases in this section:

1. The polynomial can be factored using known methods: greatest common factor and trinomial factoring.
2. The polynomial is given in factored form.
3. Technology is used to determine the intercepts.

Example 9: Finding the x-Intercepts of a Polynomial Function by Factoring

Find the x-intercepts of [latex]f\left(x\right)={x}^{6}-3{x}^{4}+2{x}^{2}[/latex].

Show Solution

We can attempt to factor this polynomial to find solutions for [latex]f\left(x\right)=0[/latex].

[latex]\begin{align} &{x}^{6}-3{x}^{4}+2{x}^{2}=0 && \\ &{x}^{2}\left({x}^{4}-3{x}^{2}+2\right)=0 && \text{Factor out the greatest common factor}. \\ &{x}^{2}\left({x}^{2}-1\right)\left({x}^{2}-2\right)=0 && \text{Factor the trinomial}. \\ &{x}^{2}\left(x+1\right)\left(x-1\right)\left({x}^{2}-2\right)=0 && \text{Factor the difference of squares}. \end{align}[/latex]

Now set each factor equal to zero and solve.

[latex]\begin{align} & {x}^{2}=0 && x+1=0 && x-1=0 && {x}^{2}-2=0 \\ &x=0 && x=-1 && x=1 && x=\pm \sqrt{2} \end{align}[/latex]

Figure 9

This gives us five x-intercepts: [latex]\left(0,0\right),\left(1,0\right),\left(-1,0\right),\left(\sqrt{2},0\right)[/latex], and [latex]\left(-\sqrt{2},0\right)[/latex]. We can see that this is an even function.

Example 10: Finding the x-Intercepts of a Polynomial Function by Factoring

Find the x-intercepts of [latex]f\left(x\right)={x}^{3}-5{x}^{2}-x+5[/latex].

Show Solution

Find solutions for [latex]f\left(x\right)=0[/latex] by factoring.

[latex]\begin{align} &{x}^{3}-5{x}^{2}-x+5=0 \\ &{x}^{2}\left(x - 5\right)-1\left(x - 5\right)=0 && \text{Factor by grouping}. \\ &\left({x}^{2}-1\right)\left(x - 5\right)=0 && \text{Factor out the common factor}. \\ &\left(x+1\right)\left(x - 1\right)\left(x - 5\right)=0 && \text{Factor the difference of squares}. \end{align}[/latex]

Now we set each factor equal to 0.

[latex]\begin{align}&x+1=0 && x - 1=0 && x - 5=0 \\ &x=-1 && x=1 && x=5 \end{align}[/latex]

Figure 10

There are three x-intercepts: [latex]\left(-1,0\right),\left(1,0\right)[/latex], and [latex]\left(5,0\right)[/latex].

Example 11: Finding the y– and x-Intercepts of a Polynomial in Factored Form

Find the y– and x-intercepts of [latex]g\left(x\right)={\left(x - 2\right)}^{2}\left(2x+3\right)[/latex].

Show Solution

The y-intercept can be found by evaluating [latex]g\left(0\right)[/latex].

[latex]g\left(0\right)={\left(0 - 2\right)}^{2}\left(2\left(0\right)+3\right)=12[/latex]

So the y-intercept is [latex]\left(0,12\right)[/latex].

The x-intercepts can be found by solving [latex]g\left(x\right)=0[/latex].

[latex]{\left(x - 2\right)}^{2}\left(2x+3\right)=0[/latex]

[latex]\begin{align}&{\left(x - 2\right)}^{2}=0 && 2x+3=0 \\ &x=2 &&x=-\frac{3}{2} \end{align}[/latex]

So the x-intercepts are [latex]\left(2,0\right)[/latex] and [latex]\left(-\frac{3}{2},0\right)[/latex].

Analysis of the Solution

We can always check that our answers are reasonable by using a graphing calculator to graph the polynomial as shown in Figure 11.

Figure 11

Example 12: Finding the x-Intercepts of a Polynomial Function Using a Graph

Find the x-intercepts of [latex]h\left(x\right)={x}^{3}+4{x}^{2}+x - 6[/latex].

Show Solution

This polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques previously discussed. Fortunately, we can use technology to find the intercepts. Keep in mind that some values make graphing difficult by hand. In these cases, we can take advantage of graphing utilities.

Looking at the graph of this function, as shown in Figure 12, it appears that there are x-intercepts at [latex]x=-3,-2[/latex], and 1.

Figure 12

We can check whether these are correct by substituting these values for x and verifying that the function is equal to 0.

Since [latex]h\left(x\right)={x}^{3}+4{x}^{2}+x - 6[/latex], we have:

[latex]h\left(-3\right)={\left(-3\right)}^{3}+4{\left(-3\right)}^{2}+\left(-3\right)-6=-27+36 - 3-6=0[/latex]

[latex]h\left(-2\right)={\left(-2\right)}^{3}+4{\left(-2\right)}^{2}+\left(-2\right)-6=-8+16 - 2-6=0[/latex]

[latex]h\left(1\right)={\left(1\right)}^{3}+4{\left(1\right)}^{2}+\left(1\right)-6=1+4+1 - 6=0[/latex]

Each x-intercept corresponds to a zero of the polynomial function and each zero yields a factor, so we can now write the polynomial in factored form.

[latex]h\left(x\right)={x}^{3}+4{x}^{2}+x - 6=\left(x+3\right)\left(x+2\right)\left(x - 1\right)[/latex]

 Determine end behavior

As we have already learned, the behavior of a graph of a polynomial function of the form

will either ultimately rise or fall as x increases without bound and will either rise or fall as x decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say –100 or –1,000.

Recall that we call this behavior the end behavior of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, [latex]{a}_{n}{x}^{n}[/latex], is an even power function, as x increases or decreases without bound, [latex]f\left(x\right)[/latex] increases without bound. When the leading term is an odd power function, as x decreases without bound, [latex]f\left(x\right)[/latex] also decreases without bound; as x increases without bound, [latex]f\left(x\right)[/latex] also increases without bound. If the leading term is negative, it will change the direction of the end behavior. The table below summarizes all four cases.

Even Degree	Odd Degree

Writing Formulas for Polynomial Functions

Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a polynomial function written in factored form will have an x-intercept where each factor is equal to zero, we can form a function that will pass through a set of x-intercepts by introducing a corresponding set of factors.

Example 13: Writing a Formula for a Polynomial Function from the Graph

Write a formula for the polynomial function shown in Figure 13.

Figure 13

Show Solution

his graph has three x-intercepts: x = –3, 2, and 5. The y-intercept is located at (0, 2). At x = –3 and x = 5, the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear. At x = 2, the graph bounces at the intercept, suggesting the corresponding factor of the polynomial will be second degree (quadratic). Together, this gives us

[latex]f\left(x\right)=a\left(x+3\right){\left(x - 2\right)}^{2}\left(x - 5\right)[/latex]

To determine the stretch factor, we utilize another point on the graph. We will use the y-intercept (0, –2), to solve for a.

[latex]\begin{align}f\left(0\right)&=a\left(0+3\right){\left(0 - 2\right)}^{2}\left(0 - 5\right) \\ -2&=a\left(0+3\right){\left(0 - 2\right)}^{2}\left(0 - 5\right) \\ -2&=-60a \\ a&=\frac{1}{30} \end{align}[/latex]

The graphed polynomial appears to represent the function [latex]f\left(x\right)=\frac{1}{30}\left(x+3\right){\left(x - 2\right)}^{2}\left(x - 5\right)[/latex].

Try It

Given the graph in Figure 14, write a formula for the function shown.

Figure 14

Show Solution

[latex]f\left(x\right)=-\frac{1}{8}{\left(x - 2\right)}^{3}{\left(x+1\right)}^{2}\left(x - 4\right)[/latex]

Using Local and Global Extrema

With quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. Even then, finding where extrema occur can still be algebraically challenging. For now, we will estimate the locations of turning points using technology to generate a graph.

Each turning point represents a local minimum or maximum. Sometimes, a turning point is the highest or lowest point on the entire graph. In these cases, we say that the turning point is a global maximum or a global minimum. These are also referred to as the absolute maximum and absolute minimum values of the function.

Local and Global Extrema

A local maximum or local minimum at x = a (sometimes called the relative maximum or minimum, respectively) is the output at the highest or lowest point on the graph in an open interval around x = a. If a function has a local maximum at a, then [latex]f\left(a\right)\ge f\left(x\right)[/latex] for all x in an open interval around x = a. If a function has a local minimum at a, then [latex]f\left(a\right)\le f\left(x\right)[/latex] for all x in an open interval around x = a.

A global maximum or global minimum is the output at the highest or lowest point of the function. If a function has a global maximum at a, then [latex]f\left(a\right)\ge f\left(x\right)[/latex] for all x. If a function has a global minimum at a, then [latex]f\left(a\right)\le f\left(x\right)[/latex] for all x.

We can see the difference between local and global extrema in Figure 15.

Figure 15

Key Equations

Key Concepts

• A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power.
• The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.
• A polynomial of degree n will have at most n x-intercepts and at most n – 1 turning points.
• The graph of a polynomial function changes direction at its turning points.
• Polynomial functions of degree 2 or more are smooth, continuous functions.
• To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.
• Another way to find the x-intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the x-axis.
• The multiplicity of a zero determines how the graph behaves at the x-intercepts, in addition to the behavior equations at each zero.
• The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.
• The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.
• The end behavior of a polynomial function depends on the leading term.

Section 4.1 Homework Exercises

1. What is the difference between an x-intercept and a zero of a polynomial function [latex]y[/latex]?

2. If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?

3. What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?

4. Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.

5. Explain how the factored form of the polynomial helps us in graphing it.

6. If the graph of a polynomial just touches the x-axis and then changes direction, what can we conclude about the factored form of the polynomial?

For the following exercises, determine whether the function is polynomial function.  If it is a polynomial function, indicate the degree and [latex]a_{n}[/latex] (leading coefficient).

7. [latex]\text{ }f\left(x\right)=6{x}^{5}[/latex]

8. [latex]\text{ }f\left(x\right)={\left({x}^{2}\right)}^{3}[/latex]

9. [latex]\text{ }f\left(x\right)=\frac{{x}^{2}}{{x}^{2}-1}[/latex]

10. [latex]\text{ }f\left(x\right)=x-{x}^{4}[/latex]

11. [latex]\text{ }f\left(x\right)=2x\left(x+2\right){\left(x - 1\right)}^{2}[/latex]

12. [latex]\text{ }f\left(x\right)={3}^{x+1}[/latex]

13. [latex]\text{ }f\left(x\right)=5\sqrt(x)+2x^{4}[/latex]

14. [latex]\text{ }f\left(x\right)=4x^{2}-\dfrac{3}{x^{3}}[/latex]

For the following exercises, find the intercepts of the functions.

15. [latex]f\left(t\right)=2\left(t - 1\right)\left(t+2\right)\left(t - 3\right)[/latex]

16. [latex]g\left(n\right)=-2\left(3n - 1\right)\left(2n+1\right)[/latex]

17. [latex]f\left(x\right)=-4(x+2)(x-2)[/latex]

18. [latex]f\left(x\right)=\left(x+3\right)\left(4{x}^{2}-1\right)[/latex]

19. [latex]f\left(x\right)=x\left({x}^{2}-2x - 8\right)[/latex]

For the following exercises, determine the least possible degree of the polynomial function shown.

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For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.

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For the following exercises, find the zeros and multiplicities.

34. [latex]f\left(x\right)={\left(x+2\right)}^{3}{\left(x - 3\right)}^{2}[/latex]

35. [latex]f\left(x\right)={x}^{2}{\left(2x+3\right)}^{3}{\left(x - 4\right)}^{2}[/latex]

36. [latex]f\left(x\right)={x}^{3}{\left(x - 1\right)}^{3}\left(x+2\right)[/latex]

37. [latex]f\left(x\right)={x}^{2}\left({x}^{2}+4x+4\right)[/latex]

38. [latex]f\left(x\right)={\left(2x+1\right)}^{3}\left(9{x}^{2}-6x+1\right)[/latex]

39. [latex]f\left(x\right)={\left(3x+2\right)}^{3}\left({x}^{2}-10x+25\right)[/latex]

40. [latex]f\left(x\right)=x\left(4{x}^{2}-12x+9\right)\left({x}^{2}+8x+16\right)[/latex]

41. [latex]f\left(x\right)={x}^{6}-{x}^{5}-2{x}^{4}[/latex]

32. [latex]f\left(x\right)=3{x}^{4}+6{x}^{3}+3{x}^{2}[/latex]

43. [latex]f\left(x\right)=4{x}^{5}-12{x}^{4}+9{x}^{3}[/latex]

44. [latex]f\left(x\right)=2{x}^{4}\left({x}^{3}-4{x}^{2}+4x\right)[/latex]

45. [latex]f\left(x\right)=4{x}^{4}\left(9{x}^{4}-12{x}^{3}+4{x}^{2}\right)[/latex]

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer.

46. The y-intercept is [latex]\left(0,-6\right)[/latex]. The x-intercepts are [latex]\left(-3,0\right),\left(2,0\right)[/latex]. Degree is 2.

End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty[/latex].

47. The y-intercept is [latex]\left(0,-4\right)[/latex]. The x-intercepts are [latex]\left(-2,0\right),\left(2,0\right)[/latex]. Degree is 2.

End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty[/latex].

48. The y-intercept is [latex]\left(0,9\right)[/latex]. The x-intercepts are [latex]\left(-3,0\right),\left(3,0\right)[/latex]. Degree is 2.

End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to -\infty ,\text{ as }x\to \infty ,f\left(x\right)\to -\infty[/latex].

49. The y-intercept is [latex]\left(0,0\right)[/latex]. The x-intercepts are [latex]\left(0,0\right),\left(2,0\right)[/latex]. Degree is 3.

End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to -\infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty[/latex].

50. The y-intercept is [latex]\left(0,1\right)[/latex]. The x-intercept is [latex]\left(1,0\right)[/latex]. Degree is 3.

End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to -\infty[/latex].

51. The y-intercept is [latex]\left(0,1\right)[/latex]. There is no x-intercept. Degree is 4.

End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty[/latex].

For the following exercises, use the graphs to write the formula for a polynomial function of least degree.

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