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Section 9.2 Polynomials

Subsection 9.2.1 Degree and Roots

In this activity, we will explore polynomial functions by building on our knowledge of power functions.

A polynomial is a function of the form

\begin{equation*} f(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \end{equation*}

where the numbers \(a_0\) through \(a_n\) are constant values, called coefficients. A polynomial is a sum of a specific set power functions (power functions with non-negative exponents).

First, we will introduce two important terms associated with polynomials: degree and root

The degree of a polynomial is the highest exponent on the variable.

Some polynomials and their degrees:

Polynomial Degree
\(-3 + 5x + x^2\) \(2\)
\(10 + 5.4x\) \(1\)
\(9x^4 + 3x\) \(4\)
\(137\) \(0\)

A root of a polynomial is a value of the input which gives an output of zero.

Test your understanding of these definitions in the next two exercises.

Now, we will see how our knowledge of first degree polynomials (a.k.a. lines) will help us to understand polynomials of higher degrees.

By now, you are already seeing the relationship between the roots of a polynomial, and the roots of its linear factors. Use this observation to make predictions about the graph of a polynomial in the next exercise.

In the next two problems, you'll predict the number of roots of a polynomial and where those roots are located.

Now, let's summarize what we've seen up to this point in the next exercise.

One important note should be made here: Not every polynomial can be written as just a product of linear factors. 1 The function \(f(x) = x^2 + 1\) is an easy example, as it has no \(x\)-intercepts. Can you think of other examples? But in this section, we will focus primarily on polynomials which can be written as a product of linear factors.

One skill we will develop here is being able to determine a formula for a polynomial if we have its graph. However, before we can do that, we need to discuss the end-behavior of a polynomial. The end-behavior of a function is what its outputs tend to do as \(x\) gets (infinitely) far from zero.

End-behavior is one thing we can tell about the graph of a polynomial, even if it is not written as a product of linear factors, such as:

\begin{gather*} y = 3x - 5x^2 + 17\\ \text{or}\\ y = -950x + x^3 \end{gather*}

Subsection 9.2.2 End Behavior

Recall that a horizontal asymptote, \(y = a\text{,}\) is a horizontal line which a function \(f(x)\) seems to resemble, as either \(x \to \infty\) or \(x \to -\infty\text{.}\)

Put another way, if the function \(f\) has a horizontal asymptote \(y = a\text{,}\) then as we evaluate \(f(x)\) at \(x\) values which are farther and farther from zero, the outputs will eventually get very close to the number \(a\text{.}\)

In calculus language, we would write

\begin{equation*} \lim_{x\to\pm\infty}f(x)=a \end{equation*}

and in words, we would say

The limit of \(f(x)\text{,}\) as \(x\) goes to positive or negative infinity, is equal to \(a\text{.}\)

In the next exercise, you will see how a horizontal asymptote of a function can be seen by a table of values.

This behavior can be seen in the graph, as the function \(f(x)\) appears to “flatten out” to the left and the right. This is an example of what is called end-behavior (or long-run behavior). From a graphical perspective, end-behavior is what the graph tends to do as the input \(x\) gets far from zero.

Figure 9.2.12. Function with horizontal asymptote \(y = 2\)
Figure 9.2.13. Function with horizontal asymptote \(y = -1\)

As you know, not every function has a horizontal asymptote. For example, power functions do not “flatten out,” but rather they either increase toward \(\infty\) (point upward) or decrease toward \(-\infty\) (point downward). So, end-behavior may be just that — the graph increasing or decreasing forever.

In the next two exercises, you will examine the end-behavior of a polynomial and learn to predict this end-behavior by observing its formula.

The rule for determining the end-behavior of a polynomial is a simple one:

A polynomial and its leading term have the same end-behavior.

And, since you already know the graphs of power functions, you should be able to predict the end-behavior of a polynomial if you know its leading term.

In the next exercise, you will get practice finding the leading term of a polynomial.

Subsection 9.2.3 The Formula for a Polynomial

We have now learned two important things we need in order to write the formula for a polynomial:

  • the relationship between the roots of a polynomial and its linear factors

  • the relationship between the end-behavior of a polynomial and its leading term

Let's put these together in order to write the formula for a polynomial.

In the next two exercises, you will be given the graph of a polynomial. Use what you know about the roots and end-behavior in order to write the formula.

Keep that last exercise in mind when finding a formula for a polynomial. You may need to control the end-behavior by flipping the graph over the \(x\)-axis.

Subsection 9.2.4 Applying Vertical Stretches

There are a few additional things we will encounter with the graph of a polynomial.

The first issue concerns using vertical stretches to the graph of a polynomial (see Chapter 6).

Imagine trying to determine the formulas for the \(3^{\rm{rd}}\) degree polynomials graphed below.

Figure 9.2.20. First Graph
Figure 9.2.21. Second Graph
Figure 9.2.22. Third Graph

They each appear to have the same end-behavior as \(x^3\text{,}\) and they all have the same \(x\)-intercepts:

\begin{equation*} (1, 0)\qquad(3, 0)\qquad(4, 0) \end{equation*}

To begin, we will give the formula the necessary linear factors:

\begin{equation*} f(x) = (x-1)(x-3)(x-4) \end{equation*}

because this gives us the correct roots and end-behavior. However, the graphs are not identical, so this formula cannot work for all of them.

Since these graphs all have the same roots of \(1\text{,}\) \(3\) and \(4\text{,}\) we might consider examining a different input value.

Fix your attention on the value \(x = 2\) for each graph.

But what about the formula for the Second Graph? Its output when \(x = 2\) is \(4\text{,}\) which is two times as large as the first graph.

This is a vertical stretch of the first graph, as you saw in Section 6.2

For the Third Graph, we see the output is \(3\) when the input is \(x = 2\text{.}\)

We may be able to tell by what factor the graph of \(f(x)\) has been vertically stretched to make this graph, but the next exercise explores an algebraic method for determining the factor.

In the next exercise, use this algebraic technique to determine the formula for a different polynomial.

Subsection 9.2.5 Multiplicity of Roots

By now, we have developed techniques to find the formulas for many kinds of polynomials if we know their roots, end-behavior, and some other point on the graph.

The last detail we will encounter involves polynomials with factors such as:

\begin{equation*} x^2 \text{ or } (x-1)^3 \text{ or } (x+6)^{10} \end{equation*}

In particular, we want to determine how having an exponent on a linear factor will affect the graph of the polynomial.

In the final exercises for this activity, you will explore these effects on the graph.

When a polynomial has a linear factor

\begin{equation*} (x - a)^k \end{equation*}

we say that the root \(x = a\) has multiplicity \(k\text{.}\)

What is important for us to know, is how the multiplicity of a root affects the graph.