Answer :
To write a polynomial function with zeros at [tex]$-5$[/tex], [tex]$-3$[/tex], and [tex]$3$[/tex], we start by using the fact that if a number [tex]$r$[/tex] is a zero of a polynomial, then [tex]$(x - r)$[/tex] is a factor of the polynomial.
1. Since the zeros are [tex]$-5$[/tex], [tex]$-3$[/tex], and [tex]$3$[/tex], the corresponding factors are:
[tex]$$
(x + 5),\quad (x + 3),\quad (x - 3).
$$[/tex]
2. The polynomial function can be written in factored form as:
[tex]$$
f(x) = (x+5)(x+3)(x-3).
$$[/tex]
3. Notice that two of the factors, [tex]$(x+3)$[/tex] and [tex]$(x-3)$[/tex], form a difference of squares. We can multiply them first:
[tex]$$
(x+3)(x-3) = x^2 - 9.
$$[/tex]
4. Now, multiply this result by the remaining factor [tex]$(x+5)$[/tex]:
[tex]$$
f(x) = (x+5)(x^2 - 9).
$$[/tex]
5. Expand the product by using the distributive property:
[tex]\[
\begin{aligned}
f(x) &= x \cdot (x^2 - 9) + 5 \cdot (x^2 - 9) \\
&= x^3 - 9x + 5x^2 - 45.
\end{aligned}
\][/tex]
6. Arrange the terms in descending order of power:
[tex]$$
f(x) = x^3 + 5x^2 - 9x - 45.
$$[/tex]
Thus, the polynomial function with zeros at [tex]$-5$[/tex], [tex]$-3$[/tex], and [tex]$3$[/tex] is:
[tex]$$
\boxed{f(x) = x^3 + 5x^2 - 9x - 45}.
$$[/tex]
1. Since the zeros are [tex]$-5$[/tex], [tex]$-3$[/tex], and [tex]$3$[/tex], the corresponding factors are:
[tex]$$
(x + 5),\quad (x + 3),\quad (x - 3).
$$[/tex]
2. The polynomial function can be written in factored form as:
[tex]$$
f(x) = (x+5)(x+3)(x-3).
$$[/tex]
3. Notice that two of the factors, [tex]$(x+3)$[/tex] and [tex]$(x-3)$[/tex], form a difference of squares. We can multiply them first:
[tex]$$
(x+3)(x-3) = x^2 - 9.
$$[/tex]
4. Now, multiply this result by the remaining factor [tex]$(x+5)$[/tex]:
[tex]$$
f(x) = (x+5)(x^2 - 9).
$$[/tex]
5. Expand the product by using the distributive property:
[tex]\[
\begin{aligned}
f(x) &= x \cdot (x^2 - 9) + 5 \cdot (x^2 - 9) \\
&= x^3 - 9x + 5x^2 - 45.
\end{aligned}
\][/tex]
6. Arrange the terms in descending order of power:
[tex]$$
f(x) = x^3 + 5x^2 - 9x - 45.
$$[/tex]
Thus, the polynomial function with zeros at [tex]$-5$[/tex], [tex]$-3$[/tex], and [tex]$3$[/tex] is:
[tex]$$
\boxed{f(x) = x^3 + 5x^2 - 9x - 45}.
$$[/tex]
