Answer :
We are given the zeros of the polynomial:
[tex]$$
x = -5, \ -5, \ 1.
$$[/tex]
This tells us that [tex]$x=-5$[/tex] is a zero of multiplicity 2, and [tex]$x=1$[/tex] is a zero of multiplicity 1. Therefore, we can write the polynomial in its factored form as:
[tex]$$
(x+5)^2 (x-1).
$$[/tex]
Now, we will expand this expression to write the polynomial in standard form.
1. First, expand the square:
[tex]$$
(x+5)^2 = x^2 + 10x + 25.
$$[/tex]
2. Next, multiply this quadratic by the linear factor [tex]$(x-1)$[/tex]:
[tex]$$
(x^2 + 10x + 25)(x - 1).
$$[/tex]
3. We apply the distributive property:
[tex]\[
\begin{aligned}
(x^2 + 10x + 25)(x - 1) &= x^2(x-1) + 10x(x-1) + 25(x-1) \\
&= (x^3 - x^2) + (10x^2 - 10x) + (25x - 25).
\end{aligned}
\][/tex]
4. Combine like terms:
- [tex]$x^3$[/tex] (only one cubic term)
- [tex]$-x^2 + 10x^2 = 9x^2$[/tex]
- [tex]$-10x + 25x = 15x$[/tex]
- The constant term is [tex]$-25$[/tex].
Putting it all together, the polynomial in standard form is:
[tex]$$
x^3 + 9x^2 + 15x - 25.
$$[/tex]
Thus, the final answer is:
[tex]$$
\boxed{x^3 + 9x^2 + 15x - 25.}
$$[/tex]
[tex]$$
x = -5, \ -5, \ 1.
$$[/tex]
This tells us that [tex]$x=-5$[/tex] is a zero of multiplicity 2, and [tex]$x=1$[/tex] is a zero of multiplicity 1. Therefore, we can write the polynomial in its factored form as:
[tex]$$
(x+5)^2 (x-1).
$$[/tex]
Now, we will expand this expression to write the polynomial in standard form.
1. First, expand the square:
[tex]$$
(x+5)^2 = x^2 + 10x + 25.
$$[/tex]
2. Next, multiply this quadratic by the linear factor [tex]$(x-1)$[/tex]:
[tex]$$
(x^2 + 10x + 25)(x - 1).
$$[/tex]
3. We apply the distributive property:
[tex]\[
\begin{aligned}
(x^2 + 10x + 25)(x - 1) &= x^2(x-1) + 10x(x-1) + 25(x-1) \\
&= (x^3 - x^2) + (10x^2 - 10x) + (25x - 25).
\end{aligned}
\][/tex]
4. Combine like terms:
- [tex]$x^3$[/tex] (only one cubic term)
- [tex]$-x^2 + 10x^2 = 9x^2$[/tex]
- [tex]$-10x + 25x = 15x$[/tex]
- The constant term is [tex]$-25$[/tex].
Putting it all together, the polynomial in standard form is:
[tex]$$
x^3 + 9x^2 + 15x - 25.
$$[/tex]
Thus, the final answer is:
[tex]$$
\boxed{x^3 + 9x^2 + 15x - 25.}
$$[/tex]
