To determine the number of [tex]\( x \)[/tex]-intercepts for the function [tex]\( f(x) = (x+5)^3(x-9)(x+1) \)[/tex], follow these steps:
1. Identify the factors of the function: The given function can be expressed as a product of simpler factors:
[tex]\[
f(x) = (x+5)^3 (x-9) (x+1)
\][/tex]
2. Set the function equal to zero: The [tex]\( x \)[/tex]-intercepts occur where the function equals zero:
[tex]\[
f(x) = 0
\][/tex]
This means:
[tex]\[
(x+5)^3 (x-9) (x+1) = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]: Determine the values of [tex]\( x \)[/tex] that make each factor equal to zero.
- For the factor [tex]\((x+5)^3\)[/tex], set [tex]\( x+5 = 0 \)[/tex]:
[tex]\[
x+5 = 0 \implies x = -5
\][/tex]
Since this factor is cubed, [tex]\( x = -5 \)[/tex] is a root with multiplicity 3.
- For the factor [tex]\((x-9)\)[/tex], set [tex]\( x-9 = 0 \)[/tex]:
[tex]\[
x-9 = 0 \implies x = 9
\][/tex]
This factor is linear, so [tex]\( x = 9 \)[/tex] is a root with multiplicity 1.
- For the factor [tex]\((x+1)\)[/tex], set [tex]\( x+1 = 0 \)[/tex]:
[tex]\[
x+1 = 0 \implies x = -1
\][/tex]
This factor is also linear, so [tex]\( x = -1 \)[/tex] is a root with multiplicity 1.
4. List the distinct roots: The distinct values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( f(x) = 0 \)[/tex] are:
[tex]\[
x = -5, \quad x = 9, \quad x = -1
\][/tex]
5. Count the number of distinct [tex]\( x \)[/tex]-intercepts: Even though [tex]\( x = -5 \)[/tex] has a multiplicity of 3, it is counted as one distinct [tex]\( x \)[/tex]-intercept.
So, the distinct [tex]\( x \)[/tex]-intercepts are:
[tex]\[
[-5, 9, -1]
\][/tex]
The number of [tex]\( x \)[/tex]-intercepts for this function is [tex]\( 3 \)[/tex].
