Let's first factor the polynomial [tex]\( P(x) = x^5 + 13x^3 \)[/tex].
### Step-by-Step Factorization:
1. Identify common factors:
- Notice that both terms, [tex]\( x^5 \)[/tex] and [tex]\( 13x^3 \)[/tex], share a common factor of [tex]\( x^3 \)[/tex].
2. Factor out the common term:
[tex]\[
x^5 + 13x^3 = x^3(x^2 + 13)
\][/tex]
Therefore, the completely factored form of the polynomial is:
[tex]\[
P(x) = x^3(x^2 + 13)
\][/tex]
### Finding the Zeros:
To find the zeros of the factors of [tex]\( P(x) \)[/tex]:
1. Set the factored form equal to zero:
[tex]\[
x^3(x^2 + 13) = 0
\][/tex]
2. Solve each factor for zero:
- For [tex]\( x^3 = 0 \)[/tex]:
[tex]\[
x = 0
\][/tex]
The zero here is [tex]\( x = 0 \)[/tex].
- For [tex]\( x^2 + 13 = 0 \)[/tex]:
[tex]\[
x^2 = -13
\][/tex]
[tex]\[
x = \pm \sqrt{-13}
\][/tex]
Since the square root of a negative number involves imaginary numbers:
[tex]\[
x = \pm \sqrt{13}i
\][/tex]
The zeros here are [tex]\( x = \sqrt{13}i \)[/tex] and [tex]\( x = -\sqrt{13}i \)[/tex].
### Determining Multiplicities:
- The factor [tex]\( x^3 \)[/tex] indicates that [tex]\( x = 0 \)[/tex] has a multiplicity of 3.
- The factor [tex]\( x^2 + 13 \)[/tex] contributes [tex]\( \sqrt{13}i \)[/tex] and [tex]\( -\sqrt{13}i \)[/tex], each with a multiplicity of 1.
### Summary:
The polynomial [tex]\( P(x) = x^5 + 13x^3 \)[/tex] can be factored and its zeros with their multiplicities are:
[tex]\[ P(x) = x^3(x^2 + 13) \][/tex]
- Zero with multiplicity 3: [tex]\( x = 0 \)[/tex]
- Zero with multiplicity 1: [tex]\( x = \sqrt{13}i \)[/tex]
- Zero with multiplicity 1: [tex]\( x = -\sqrt{13}i \)[/tex]
