To determine the zeros of the polynomial [tex]\( f(x) = 3(x+5)^3(x-5)^2(x+7)(x-4)^3 \)[/tex] and their corresponding multiplicities, we need to examine each factor of the polynomial and their respective exponents.
1. Zeros of multiplicity one:
- We look for factors raised to the power of 1. The factor [tex]\( (x+7) \)[/tex] is raised to the power of 1.
- The zero corresponding to this factor is [tex]\( x = -7 \)[/tex].
Therefore, the zero of multiplicity one is:
[tex]\[
7
\][/tex]
2. Zeros of multiplicity two:
- We search for factors raised to the power of 2. The factor [tex]\( (x-5) \)[/tex] is raised to the power of 2.
- The zero corresponding to this factor is [tex]\( x = -5 \)[/tex].
Therefore, the zero of multiplicity two is:
[tex]\[
-5
\][/tex]
3. Zeros of multiplicity three:
- We identify factors raised to the power of 3. The factors [tex]\( (x+5) \)[/tex] and [tex]\( (x-4) \)[/tex] are each raised to the power of 3.
- The zeros corresponding to these factors are [tex]\( x = 5 \)[/tex] and [tex]\( x = -4 \)[/tex].
Therefore, the zeros of multiplicity three are:
[tex]\[
5, -4
\][/tex]
To summarize:
- The zero of multiplicity one is: [tex]\( 7 \)[/tex]
- The zero of multiplicity two is: [tex]\( -5 \)[/tex]
- The zeros of multiplicity three are: [tex]\( 5, -4 \)[/tex]
