To find the zeros of the polynomial function [tex]\( f(x) = x^3 + 9x^2 - 16x - 144 \)[/tex] and determine their multiplicities, we need to factor the polynomial or use techniques such as the Rational Root Theorem and synthetic division.
Let's begin by finding possible rational zeros using the Rational Root Theorem. The Rational Root Theorem states that any rational root of the polynomial [tex]\( f(x) \)[/tex] must be a factor of the constant term (-144) divided by a factor of the leading coefficient (1), which means possible rational roots are the factors of -144.
The factors of -144 are:
[tex]\[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 9, \pm 12, \pm 16, \pm 18, \pm 24, \pm 36, \pm 48, \pm 72, \pm 144. \][/tex]
We start by testing these potential roots in the polynomial function to see which ones are actual zeros.
First, test [tex]\( x = -9 \)[/tex]:
[tex]\[
f(-9) = (-9)^3 + 9(-9)^2 - 16(-9) - 144
= -729 + 729 + 144 - 144
= 0
\][/tex]
Since [tex]\( f(-9) = 0 \)[/tex], [tex]\( x = -9 \)[/tex] is a zero of the polynomial [tex]\( f(x) = x^3 + 9x^2 - 16x - 144 \)[/tex].
Now we perform synthetic division to factor [tex]\( x + 9 \)[/tex] out of the polynomial:
[tex]\[
\begin{array}{r|rrrr}
-9 & 1 & 9 & -16 & -144 \\
& & -9 & 0 & 144 \\
\hline
& 1 & 0 & -16 & 0 \\
\end{array}
\][/tex]
The synthetic division gives us the quotient [tex]\(x^2 - 16\)[/tex]. Thus, we can factor [tex]\( f(x) \)[/tex] as:
[tex]\[
f(x) = (x + 9)(x^2 - 16)
\][/tex]
Next, factor [tex]\( x^2 - 16 \)[/tex]:
[tex]\[
x^2 - 16 = (x + 4)(x - 4)
\][/tex]
Therefore, the fully factored form of [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = (x + 9)(x + 4)(x - 4)
\][/tex]
From the factored form, we can identify the zeros of [tex]\( f(x) \)[/tex]:
- [tex]\( x = -9 \)[/tex]
- [tex]\( x = -4 \)[/tex]
- [tex]\( x = 4 \)[/tex]
The multiplicity of each zero is 1, because each factor appears only once in the factored form of the polynomial.
Now classify the behavior at each zero:
- Since each zero has a multiplicity of 1, the graph of the polynomial function will cross the [tex]\( x \)[/tex]-axis at each zero.
Thus, we have three zeros with the following properties:
- Zero at [tex]\( x = -9 \)[/tex] with multiplicity 1
- Zero at [tex]\( x = -4 \)[/tex] with multiplicity 1
- Zero at [tex]\( x = 4 \)[/tex] with multiplicity 1
Therefore, the correct choice is:
C. There are three zeros.
- The multiplicity of the smallest zero is [tex]\( 1 \)[/tex] (for [tex]\( x = -9 \)[/tex]).
- The multiplicity of the largest zero is [tex]\( 1 \)[/tex] (for [tex]\( x = 4 \)[/tex]).
- The multiplicity of the other zero is [tex]\( 1 \)[/tex] (for [tex]\( x = -4 \)[/tex]).
