To form a polynomial \( f(x) \) with real coefficients, given the degree and zeros, we need to consider the following steps:
1. Write down the given zeros: The zeros provided are \( -5 - 2i \) and \( -3 \) with a multiplicity of 2.
2. Include the complex conjugate: Since the polynomial must have real coefficients, the complex zeros must occur in conjugate pairs. Thus, if \( -5 - 2i \) is a zero, its complex conjugate \( -5 + 2i \) must also be a zero.
3. List all zeros: The zeros of the polynomial will be \( -5 - 2i, -5 + 2i, -3, -3 \).
4. Construct factors from zeros: Each zero \( \alpha \) will correspond to a factor \( (x - \alpha) \). Thus, the factors will be:
- For \( -5 - 2i \): \( (x - (-5 - 2i)) = (x + 5 + 2i) \)
- For \( -5 + 2i \): \( (x - (-5 + 2i)) = (x + 5 - 2i) \)
- For \( -3 \) (with multiplicity 2): \( (x - (-3))^2 = (x + 3)^2 \)
5. Form the polynomial: Multiply the factors to get the polynomial:
[tex]\[
f(x) = (x + 5 + 2i)(x + 5 - 2i)(x + 3)^2
\][/tex]
6. Multiply conjugate pairs: The product of the conjugate pairs will be a quadratic polynomial with real coefficients.
[tex]\[
(x + 5 + 2i)(x + 5 - 2i) = (x + 5)^2 - (2i)^2 = (x + 5)^2 - 4(-1) = (x + 5)^2 + 4 = (x + 5)^2 + 4
\][/tex]
Expand:
[tex]\[
(x + 5)^2 + 4 = x^2 + 10x + 25 + 4 = x^2 + 10x + 29
\][/tex]
7. Combine with the quadratic term from the repeated zero \( (-3) \):
[tex]\[
f(x) = (x^2 + 10x + 29)(x + 3)^2
\][/tex]
Expand \( (x + 3)^2 \) first:
[tex]\[
(x + 3)^2 = x^2 + 6x + 9
\][/tex]
8. Expand the entire polynomial by multiplying \( (x^2 + 10x + 29) \) with \( (x^2 + 6x + 9) \):
[tex]\[
f(x) = (x^2 + 10x + 29)(x^2 + 6x + 9)
\][/tex]
Use distributive property to expand (multiply each term in the first polynomial by each term in the second polynomial):
[tex]\[
\begin{align*}
f(x) &= (x^2)(x^2) + (x^2)(6x) + (x^2)(9) \\
&+ (10x)(x^2) + (10x)(6x) + (10x)(9) \\
&+ (29)(x^2) + (29)(6x) + (29)(9)
\\
&= x^4 + 6x^3 + 9x^2 \\
&+ 10x^3 + 60x^2 + 90x \\
&+ 29x^2 + 174x + 261
\end{align*}
\][/tex]
9. Combine like terms:
[tex]\[
f(x) = x^4 + (6x^3 + 10x^3) + (9x^2 + 60x^2 + 29x^2) + (90x + 174x) + 261
\][/tex]
Simplify:
[tex]\[
f(x) = x^4 + 16x^3 + 98x^2 + 264x + 261
\][/tex]
Thus, the polynomial \( f(x) \) with real coefficients, having degree 4, and zeros \( -5 - 2i, -5 + 2i, -3, -3 \), is:
[tex]\[
f(x) = x^4 + 16x^3 + 98x^2 + 264x + 261
\][/tex]
