To determine how many different roots the polynomial function [tex]\( y = (x + 4)(x - 2)^2(x + 7) \)[/tex] has, we need to find the values of [tex]\( x \)[/tex] that make [tex]\( y = 0 \)[/tex]. These values are known as the roots of the polynomial.
First, let's set the polynomial equal to zero:
[tex]\[ (x + 4)(x - 2)^2(x + 7) = 0 \][/tex]
For the product of several factors to be zero, at least one of the factors must be zero. We will set each factor equal to zero and solve for [tex]\( x \)[/tex]:
1. [tex]\( x + 4 = 0 \)[/tex]
[tex]\[ x = -4 \][/tex]
2. [tex]\( (x - 2)^2 = 0 \)[/tex]
[tex]\[ x - 2 = 0 \][/tex]
[tex]\[ x = 2 \][/tex]
Note that [tex]\( x = 2 \)[/tex] is a root with multiplicity 2 because the factor [tex]\((x - 2)\)[/tex] is squared.
3. [tex]\( x + 7 = 0 \)[/tex]
[tex]\[ x = -7 \][/tex]
We have found the roots:
- [tex]\( x = -4 \)[/tex]
- [tex]\( x = 2 \)[/tex] (with multiplicity 2)
- [tex]\( x = -7 \)[/tex]
Despite one of the roots ([tex]\( x = 2 \)[/tex]) having a multiplicity of 2, it still counts as a single distinct root.
Now, we count the number of different roots:
- [tex]\( -4 \)[/tex]
- [tex]\( 2 \)[/tex]
- [tex]\( -7 \)[/tex]
There are 3 different roots.
Therefore, the correct answer is:
A. 3
