To form a polynomial with the given zeros and their multiplicities, we follow these steps:
1. Start by writing the factors of the polynomial based on the given zeros. In this case, the zeros are:
- [tex]\( -5 \)[/tex] with a multiplicity of 1, so the factor is [tex]\( (x + 5) \)[/tex].
- [tex]\( -1 \)[/tex] with a multiplicity of 2, so the factor is [tex]\( (x + 1)^2 \)[/tex].
2. Multiply these factors to form the polynomial:
[tex]\[
f(x) = (x + 5) \cdot (x + 1)^2
\][/tex]
3. Next, expand the polynomial to get it in standard form. We will do this step-by-step.
First, expand [tex]\( (x + 1)^2 \)[/tex]:
[tex]\[
(x + 1)^2 = (x + 1)(x + 1) = x^2 + 2x + 1
\][/tex]
Now, multiply [tex]\( (x + 5) \)[/tex] by [tex]\( x^2 + 2x + 1 \)[/tex]:
[tex]\[
f(x) = (x + 5)(x^2 + 2x + 1)
\][/tex]
Distribute [tex]\( (x + 5) \)[/tex] across the terms inside the parentheses:
[tex]\[
\begin{aligned}
f(x) &= x(x^2 + 2x + 1) + 5(x^2 + 2x + 1) \\
&= (x^3 + 2x^2 + x) + (5x^2 + 10x + 5) \\
\end{aligned}
\][/tex]
4. Combine like terms to simplify the expression:
[tex]\[
\begin{aligned}
f(x) &= x^3 + (2x^2 + 5x^2) + (x + 10x) + 5 \\
&= x^3 + 7x^2 + 11x + 5
\end{aligned}
\][/tex]
Thus, the polynomial with integer coefficients and a leading coefficient of 1, given the zeros and their multiplicities, is:
[tex]\[
f(x) = x^3 + 7x^2 + 11x + 5
\][/tex]
