To find the polynomial function [tex]\( f(x) \)[/tex] with real coefficients that has zeros at [tex]\(-3\)[/tex] and [tex]\(-1\)[/tex] with each having a multiplicity of 2, follow these steps:
1. Identify the roots and their multiplicities:
The roots are [tex]\( -3 \)[/tex] and [tex]\( -1 \)[/tex] each with multiplicity 2.
2. Write the factors corresponding to these roots:
Since the root [tex]\( -3 \)[/tex] has a multiplicity of 2, it corresponds to the factor [tex]\( (x + 3) \)[/tex] squared.
Similarly, the root [tex]\( -1 \)[/tex] with multiplicity 2 corresponds to the factor [tex]\( (x + 1) \)[/tex] squared.
Therefore, the polynomial can be written as:
[tex]\[
f(x) = (x + 3)^2 \cdot (x + 1)^2
\][/tex]
3. Expand the polynomial:
First, expand each squared term:
[tex]\[
(x + 3)^2 = x^2 + 6x + 9
\][/tex]
[tex]\[
(x + 1)^2 = x^2 + 2x + 1
\][/tex]
4. Multiply the expanded terms:
Now, multiply [tex]\( (x^2 + 6x + 9) \)[/tex] and [tex]\( (x^2 + 2x + 1) \)[/tex]:
[tex]\[
f(x) = (x^2 + 6x + 9)(x^2 + 2x + 1)
\][/tex]
5. Perform the multiplication:
Use the distributive property (FOIL method) for binomials:
[tex]\[
\begin{align*}
f(x) = (x^2 + 6x + 9)(x^2 + 2x + 1) &= x^2(x^2 + 2x + 1) + 6x(x^2 + 2x + 1) + 9(x^2 + 2x + 1) \\
&= x^4 + 2x^3 + x^2 + 6x^3 + 12x^2 + 6x + 9x^2 + 18x + 9
\end{align*}
\][/tex]
6. Combine like terms:
[tex]\[
f(x) = x^4 + (2x^3 + 6x^3) + (x^2 + 12x^2 + 9x^2) + (6x + 18x) + 9
\][/tex]
[tex]\[
f(x) = x^4 + 8x^3 + 22x^2 + 24x + 9
\][/tex]
Thus, the polynomial function is:
[tex]\[
f(x) = x^4 + 8x^3 + 22x^2 + 24x + 9
\][/tex]
Among the given choices, the correct equation is:
[tex]\[
f(x) = x^4 + 8x^3 + 22x^2 + 24x + 9
\][/tex]
So, the answer is:
[tex]\[ \boxed{x^4 + 8x^3 + 22x^2 + 24x + 9} \][/tex]
