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Section 8.2 Short-Run Behavior

Last section, we categorized polynomials by the long-run behavior. We now take a closer look at how these functions behave for smaller \(x\)-values. In particular, we see how the polynomial behaves around its roots.
In this section, we will pay close attention to the factors of polynomials. Just like we factored quadratic functions, we can factor polynomials of higher degrees. A polynomial is in factored form if it looks like \(f(x)=f_1(x) \cdot f_2(x) \cdot \ldots \cdot f_k(x)\) and each of the factors \(f_1(x), f_2(x), \ldots, f_3(x)\) cannot be factored further.

Example 8.13.

Factor the polynomial \(x^3 - 2x^2 + x\text{.}\)
Solution.
We can first factor out an \(x\) from each term to obtain
\begin{align*} x^3 - 2x^2 + x \amp= x(x^2 - 2x + 1). \end{align*}
Then, we factor the quadratic part and get
\begin{gather*} x(x^2 -2x +1) = x(x-1)(x-1). \end{gather*}
So our fully factored polynomial is
\begin{gather*} x^3 - 2x^2 + x = x(x-1)^2. \end{gather*}

Subsection 8.2.1 \(x\)-Intercepts (roots) and Short-Run Behavior

We have seen before that the \(x\)-intercepts of a quadratic polynomial, \(f(x) = ax^2 + bx + c\text{,}\) occur at values of \(x\) for which \(f(x) = 0\text{,}\) that is, at the real-valued solutions of the equation \(ax^2 + bx + c = 0\text{.}\) The same holds true for polynomials of higher degree.
Solutions of the equation \(P(x) = 0\) are called zeros of the polynomial \(P\text{.}\) In Example 8.7, we graphed the cubic polynomial \(P(x) = x^3 - 4x\text{.}\) Its \(x\)-intercepts are the solutions of the equation \(x^3 - 4x = 0\text{,}\) which we can solve by factoring the polynomial \(P(x)\text{.}\)
\begin{align*} x^3 - 4x \amp= 0 \\ x(x - 2)(x + 2) \amp= 0 \end{align*}
The zeros of \(P\) are \(0\text{,}\) \(2\text{,}\) and \(-2\text{.}\) Each zero of \(P\) corresponds to a linear factor of \(P(x)\text{.}\) This result suggests the following theorem, which holds for any polynomial \(P\text{.}\)

Factor Theorem.

Let \(P(x)\) be a polynomial with real number coefficients. Then \((x - a)\) is a factor of \(P(x)\) if and only if \(P(a) = 0\text{.}\)
Because a polynomial function of degree \(n\) can have at most \(n\) linear factors of the form \((x - a)\text{,}\) it follows that \(P\) can have at most \(n\) distinct zeros. If a zero appears more than once, we describe the number of times it appears as the multiplicity.
Another way of saying this is that if \(P(x)\) is a polynomial of degree \(n\text{,}\) the equation \(P(x) = 0\) can have at most \(n\) distinct solutions.
Not every function will factor into linear factors. If the there are non-linear factors after the function is fully factored, some solutions will be complex numbers. For example, the function \(x^3 + x = x(x^2 + 1)\) has one linear factor, \(x \text{,}\) and one non-linear factor, \(x^2 + 1 \text{,}\) which can’t be factored further, so the only real solution is \(x = 0\) (the other two solutions are complex numbers). We do not cover complex numbers in this course but there are plenty of reasonable resources available online if you wish to know more.
Because only real-valued solutions appear on the graph as \(x\)-intercepts, we have the following corollary to the factor theorem.

\(x\)-Intercepts of Polynomials.

A polynomial of degree \(n\) can have at most \(n\) \(x\)-intercepts.
If some of the zeros of \(P\) are complex numbers, they will not appear on the graph, so a polynomial of degree \(n\) may have fewer than \(n\) \(x\)-intercepts.

Example 8.14.

Find the real-valued zeros of each of the following polynomials, and list the \(x\)-intercepts of its graph.
  1. \(\displaystyle f(x) = x^3 + 6x^2 + 9x \)
  2. \(\displaystyle g(x) = x^4 - 3x^2 - 4 \)
Solution.
  1. Factor the polynomial to obtain
    \begin{align*} f(x) \amp= x(x^2 + 6x + 9) \\ \amp= x(x + 3)(x + 3). \end{align*}
    By the factor theorem, the zeros of \(f\) are \(0\text{,}\) \(-3\text{,}\) and \(-3\text{.}\) (We say that \(f\) has a zero of multiplicity two at \(-3\text{.}\)) Because all of these are real numbers, all will appear as \(x\)-intercepts on the graph. Thus, the \(x\)-intercepts occur at \((0, 0)\) and at \((-3, 0)\text{.}\)
  2. Factor the polynomial to obtain
    \begin{align*} g(x) \amp= (x^2 - 4)(x^2 + 1) \\ \amp= (x - 2)(x + 2)(x^2 + 1). \end{align*}
    Because \(x^2 + 1\) cannot be factored in real numbers, the graph has only two \(x\)-intercepts, at \((-2, 0)\) and \((2, 0)\text{.}\) Hence, \(g(x)\) has only \(x=-2\) and \(x=2\) as its zeros. The graphs of both polynomials are shown in Figure 8.15.
    described in detail following the image
    a cubic with two zeros and a quartic with two zeros
    Figure 8.15.

Example 8.16.

  1. Find the real-valued zeros of \(P(x) = -x^4 + x^3 + 2x^2\) by factoring.
  2. Sketch a rough graph of \(y = P(x)\) by hand.
Solution.
  1. Factor the polynomial to obtain
    \begin{equation*} \begin{aligned} P(x) \amp= -x^4+x^3+2x^2 \\ \amp= -x^2(x-2)(x+1) \end{aligned} \end{equation*}
    By the factor theorm, the zeros of \(P\) are \(0\text{,}\) \(2\text{,}\) and \(-1\text{.}\)
  2. described in detail following the image
    quartic

Subsection 8.2.2 Zeros of Multiplicity Two or Three

The appearance of the graph near an \(x\)-intercept is determined by the multiplicity of the zero there. For example, both real zeros of the polynomial \(g(x) = x^4 - 3x^2 - 4\) in Example 8.14b are of multiplicity one, and the graph crosses the \(x\)-axis at each intercept.
However, the polynomial \(f(x) = x^3 + 6x^2 + 9x\) in Example 8.14a has a zero of multiplicity two at \(x = -3\text{.}\) The graph of \(f\) just touches the \(x\)-axis and then reverses direction without crossing the axis.
To understand what happens in general, compare the graphs of the three polynomials in Figure 8.17.
described in detail following the image
plots of a linear, quadratic, and cubic, each with a single zero
Figure 8.17.
  1. In Figure 8.17a, \(L(x) = x - 2\) has a zero of multiplicity one at \(x = 2\text{,}\) and its graph crosses the \(x\)-axis there.
  2. In Figure 8.17b, \(Q(x) = (x - 2)^2\) has a zero of multiplicity two at \(x = 2\text{,}\) and its graph touches the \(x\)-axis there but changes direction without crossing.
  3. In Figure 8.17c, \(C(x) = (x - 2)^3\) has a zero of multiplicity three at \(x = 2\text{.}\) In this case, the graph makes an S-shaped curve at the intercept, like the graph of \(y = x^3\text{.}\)
Near its \(x\)-intercepts, the graph of a polynomial takes one of the characteristic shapes illustrated in Figure 8.17.

Multiplicities of Roots.

Let \(f(x) \) be a polynomial function.
  1. At a root of even multiplicity the graph of \(f(x)\) bounces off the \(x\)-axis.
  2. At a root of odd multiplicity the graph of \(f(x)\) crosses the \(x\)-axis.

Note 8.18.

Although we will not consider zeros of multiplicity greater than three, they correspond to similar behavior in the graph: At a zero of odd multiplicity, the graph has an inflection point at the intercept; its graph makes an S-shaped curve. At a zero of even multiplicity, the graph has a turning point; it changes direction without crossing the \(x\)-axis.

Example 8.19.

Graph the polynomial
\begin{equation*} f(x) = (x + 2)^3(x - 1)(x - 3)^2. \end{equation*}
Solution.
The polynomial has degree six, an even number, so its graph starts at the upper left and extends to the upper right. The \(y\)-coordinate of the \(y\)-intercept is
\begin{equation*} f(0) = (2)^3(-1)(-3)^2 = -72. \end{equation*}
Note that \(f\) has a zero of multiplicity three at \(x = -2\text{,}\) a zero of multiplicity one at \(x = 1\text{,}\) and a zero of multiplicity two at \(x = 3\text{.}\) The graph has an S-shaped curve at \(x = -2\text{,}\) crosses the \(x\)-axis at \(x = 1\text{,}\) touches the \(x\)-axis at \(x = 3\text{,}\) and then changes direction, as shown in Figure 8.20.
described in detail following the image
graph of sixth degree polynomial with zero of multiplicity 3, another of multiplicity two. and a zero of multiplicity 1
Figure 8.20.

Exercises 8.2.3 Exercises

1. Zeros and Long-Run Behavior.

Find the zeros and \(y\)-intercepts of the following polynomials. Select all statements that can be used to describe the long-run behavior for each.
a) \(y=-2 x^2 + 10 x\)
Zeros:
\(y\)-intercept: \((\) , \()\)
Long-run behavior:
  • As \(x\) goes to positive infinity, the \(y\)-values of the polynomial go to negative infinity.
  • As \(x\) goes to positive infinity, the \(y\)-values of the polynomial also go to positive infinity.
  • As \(x\) goes to negative infinity, the \(y\)-values of the polynomial go to positive infinity.
  • As \(x\) goes to negative infinity, the \(y\)-values of the polynomial also go to negative infinity.
b) \(y=4 x^2 - 36\)
Zeros:
\(y\)-intercept: \((\) , \()\)
Long-run behavior:
  • As \(x\) goes to positive infinity, the \(y\)-values of the polynomial go to negative infinity.
  • As \(x\) goes to negative infinity, the \(y\)-values of the polynomial go to positive infinity.
  • As \(x\) goes to positive infinity, the \(y\)-values of the polynomial also go to positive infinity.
  • As \(x\) goes to negative infinity, the \(y\)-values of the polynomial also go to negative infinity.
Hint.
a) Recall the zeros are where the function’s graph crosses the horizontal axis. In other words, where the output is 0. So we can find this information by setting the function equal to 0. It may be useful to factor the function completely to answer this question.
We evaluate this function at \(x=0\) to find the \(y\)-intercept.
For the long-run behavior of this function, consider the long-run behavior of the leading term. In this case, the polynomial is a downward facing parabola. Think about what that means as \(x\) goes to \(\infty\) and \(-\infty\text{.}\)
b) You can use the same process as in part a) here.

2. Zeros of a Polynomial.

Using an online tool, determine the number of zeros (i.e., the number of times the function makes contact with the \(x\)-axis) of the polynomial \(f(x)=x^5 + 2.8 x^4 + 5.96 x^3 + 11.2 x^2 + 7.84 x\text{.}\) Again using this online tool, estimate these zeros. Is the multiplicity of each zero even or odd?
Number of zeros:
Estimation and multiplicity (select all that apply):
  • \((1.4, 0)\text{,}\) even multiplicity
  • \((-2, 0)\text{,}\) even multiplicity
  • \((0, 0)\text{,}\) even multiplicity
  • \((0, 0)\text{,}\) odd multiplicity
  • \((-1.4, 0)\text{,}\) odd multiplicity
  • \((-1.4, 0)\text{,}\) even multiplicity
Hint.
Use the graph to help you find the zeros (\(x\)-intercepts) of the function. Remember the zeros of a function occur when the output is 0.
A zero with even multiplicity will look like it bounces off the \(x\)-axis on the graph. A zero with odd multiplicity will cross the \(x\)-axis on the graph.

3. Properties of Polynomials.

In this problem we will examine the short-run behavior of the following functions.
a) \(f(x)=(x+3)^2 (x-2)(x-4)^2\)
b) \(g(x)=(x+1)(x-2)^3\)
c) \(m(x)=-(x+6)(x+3)(x-1)(x-2)\)
d) \(n(x)= (x-4)(x+2)^2 (x-3)^3\)
Complete the following table using the formulas for each of the above functions and any additional information you can obtain from the graph of each function. You may want to sketch graphs on your own piece of paper. You are welcome to use an online graphing utility like www.desmos.com if you would like
function leading term degree number of (distinct) zeros
\(f(x)\)
\(g(x)\) \(2\)
\(m(x)\) \(4\)
\(n(x)\)
Function features
b) In the order the roots appear in the factored forms of the functions provided, list the roots, the multiplicity of the roots, and if the function "bounces" off or "crosses" the \(x\)-axis at that root. If there are more blanks provided than needed for a given function, enter NONE in all remaining, unneeded blanks. Most of the information for function \(f(x)\) has been completed as an example.
\(f(x)\text{:}\) \(x = -3\text{,}\) multiplicity: \(2\text{,}\)
\(\hspace{2mm}\)\(x = 2\text{,}\) multiplicity: \(1\text{,}\)
\(\hspace{2mm}\)\(x =4\text{,}\) multiplicity: \(2\text{,}\)
\(\hspace{2mm}\)\(x =\)NONE, multiplicity: NONE, NONE
\(g(x)\text{:}\) \(x =\) , multiplicity: ,
\(\hspace{2mm}\)\(x =\) , multiplicity: ,
\(\hspace{2mm}\) \(x =\) , multiplicity: ,
\(\hspace{2mm}\)\(x =\) , multiplicity: ,
\(m(x)\text{,}\) \(x =\) , multiplicity: ,
\(\hspace{2mm}\) \(x =\) , multiplicity: ,
\(\hspace{2mm}\) \(x =\) , multiplicity: ,
\(\hspace{2mm}\) \(x =\) , multiplicity: ,
\(n(x)\text{,}\) \(x =\) , multiplicity: ,
\(\hspace{2mm}\)\(x =\) , multiplicity: ,
\(\hspace{2mm}\) \(x =\) , multiplicity: ,
\(\hspace{2mm}\) \(x =\) , multiplicity: ,
Hint.
Recall the leading term is the term in the polynomial with the largest valued exponent. The degree of a polynomial is the value of the largest exponent (it’s the exponent on the leading term).
The number of distinct zeros is given by the number of distinct factors. For example, \(f(x)=x^2(x-1)\) has 2 distinct zeros.
The multiplicity of a zero is given by the exponent of the factor. For example, the zero \(x=0\) has multiplicity 2 in \(f(x)\) above. From looking at the graph of the function, you can determine if the function bounces off the \(x\)-axis or crosses the \(x\)-axis at each given zero.

4. Zeros and Long-Run Behavior.

a) Let \(k(x) = 2 (x + 5)(x^2-16)\text{.}\) What are the zeros of \(k\text{?}\) The \(y\)-intercept? Enter the zeros in order from smallest to largest. If there are more blanks provided than you need for the zeros, enter "NONE" for the remaining ordered pairs.
Zeros: \((\) , \()\text{,}\) \((\) , \()\text{,}\) \((\), \()\)
\(y\)-intercept: \((\), \()\)
b) Which statements describe the long-run behavior of \(k(x)\text{?}\) Select all that apply.
  • \(\displaystyle \displaystyle \lim_{x \to -\infty} k(x) = \infty\)
  • \(\displaystyle \displaystyle \lim_{x \to -\infty} k(x) = -\infty\)
  • \(\displaystyle \displaystyle \lim_{x \to \infty} k(x) = \infty\)
  • \(\displaystyle \displaystyle \lim_{x \to \infty} k(x) = -\infty\)
Hint.
a) To find all zeros, it may be useful to factor the function completely. Recall the zeros occur when the output is 0, so you can set the equation equal to 0.
The \(y\)-intercept occurs when \(x=0\text{,}\) so you should evaluate the function at \(x=0\text{.}\)
b) Recall the long-run behavior of a polynomial is the same as the long-run behavior of its leading term. Find the leading term of the polynomial to determine the long-run behavior.

5. Formula from Graph.

Give a formula for the polynomial of least degree pictured here.
\(f(x) =\)
Hint.
Find the zeros on the graph and determine if they should have even or odd multiplicity (recall odd multiplicity means the function will cross the \(x\)-axis at the \(x\)-intercept and even multiplicity means the function will bounce off the \(x\)-axis at the \(x\)-intercept). We want the least degree polynomial, so you should choose multiplicity 1 (for odd multiplicities) or 2 (for even multiplicities).
Write a formula in factored form:
\begin{equation*} y=a(x-r_1)^{p_1}(x-r_2)^{p_2}(x-r_3)^{p_3}, \end{equation*}
where \(a\) is some stretch factor. Use another point on the graph (the \(y\)-intercept is a good choice) to help you find the value of \(a\text{.}\)

6. Finding a Formula.

Select a possible formula for the polynomial \(f(x)\text{,}\) given that \(f(x)\) has degree three, \(f(-2)=f(2)=f(6)=0\text{,}\) and \(\lim_{x\to \infty}f(x)=-\infty\text{.}\)
  • \(\displaystyle -(x + 2)^2 (x-2)(x-6)\)
  • \(\displaystyle -(x + 2)(x-2)(x-6)\)
  • \(\displaystyle (x + 2)(x + 2)(x-6)\)
  • \(\displaystyle -(x + 2)(x + 6)(x + 2)\)
Is there more than one possible solution?
  • No, because we know the three roots and it is a degree three polynomial.
  • Yes, because there could be different multiplicities for the roots.
  • No, because there is only one function that meets all of the requirements.
  • Yes, because the leading coefficient could be any negative number.
Hint.
Determine what \(f(-2)=0\) means in this context/what information that gives you. Try to write an equation in factored form for the function \(f(x)\text{.}\)
Think about what must be true about the leading term/coefficient to fulfill the long-run behavior given.
For part b) Think about what would change if you applied a vertical stretch or compression.

7. Identifying Graphs.

Without using a calculator or online source, select the graph of the polynomial \(y = 3 x^4 - 12 x^2\text{.}\)
A B C D
Hint.
Try to find the \(x\)-intercepts of the given function and the long-run behavior. It may be useful to factor the function completely to see what the multiplicities are (to determine if the function should bounce or cross at a given \(x\)-intercept).

8. Application: Volume of a Box.

A company wants to design a box with four rectangular sides and a square top and bottom.
a) Letting \(x\) denote the length of one of the sides of the bottom of the box, and letting \(y\) denote the height of the box, sketch a picture of the box, labeling the length, width, and height. The company wants to use exactly 121 cm\(^{2}\) of material for the box. That is, the box’s surface area will be equal to 121 cm\(^{2}\text{.}\) Write an equation in terms of \(x\) and \(y\) representing this constraint, putting the variables on the left side.
\(=\) cm\(^{2}\)
Solve for \(y\) to find \(y\) as a function of \(x\text{.}\)
\(y=\)
b) Using your work from part (a), write a formula for the volume of the box, \(V(x)\text{,}\) as a function of \(x\text{.}\) Simplify as much as possible.
\(V(x)=\)
c) Use an online source to approximate the maximum value of \(V(x)\) on the interval \([0,10]\text{.}\) What \(x\) value corresponds to this maximum value?
What does this mean in the context of the problem?
  • This is the length of the side of the box that produces the maximum volume of a box with a surface area of 121 cm\(^2\text{.}\)
  • This is the maximum volume of a box with a surface area of 121 cm\(^2\text{.}\)
  • This is the area of the base of the box that produces the maximum volume of a box with a surface area of 121 cm\(^2\text{.}\)
Hint.
a) To find the surface area of a box, we want to add up the area of each side. Use the given information to find the area of each side of the box and add them all together. This expression should be set equal to the given surface area.
Rewrite the equation to get \(y\) by itself.
b) Try to write an equation for the volume of the box, using \(x\) and \(y). Once you have that equation, plug in the equation from part a) for \(y\text{.}\)
c) Graph the equation from part b) (we recommend using Desmos). Use the graph to find the appropriate value of \(x\) in the given domain. Think about what the units are for the value of \(x\) to give the right interpretation.
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