Answer :
Sure! Let's use the Factor Theorem to determine which polynomial function has the zeros [tex]\(4\)[/tex], [tex]\(\sqrt{7}\)[/tex], and [tex]\(-\sqrt{7}\)[/tex].
The Factor Theorem states that if [tex]\(c\)[/tex] is a zero of a polynomial [tex]\(f(x)\)[/tex], then [tex]\((x - c)\)[/tex] is a factor of [tex]\(f(x)\)[/tex].
Given the zeros are [tex]\(4\)[/tex], [tex]\(\sqrt{7}\)[/tex], and [tex]\(-\sqrt{7}\)[/tex], the polynomial can be expressed as:
[tex]\[ f(x) = (x - 4)(x - \sqrt{7})(x + \sqrt{7}) \][/tex]
First, let's simplify the factors involving the square roots:
[tex]\[ (x - \sqrt{7})(x + \sqrt{7}) = x^2 - (\sqrt{7})^2 = x^2 - 7 \][/tex]
So, the polynomial becomes:
[tex]\[ f(x) = (x - 4)(x^2 - 7) \][/tex]
Next, we need to expand this product:
[tex]\[
\begin{align*}
f(x) & = (x - 4)(x^2 - 7) \\
& = x \cdot (x^2 - 7) - 4 \cdot (x^2 - 7) \\
& = x^3 - 7x - 4x^2 + 28 \\
& = x^3 - 4x^2 - 7x + 28
\end{align*}
\][/tex]
Therefore, the polynomial function [tex]\(f(x)\)[/tex] is:
[tex]\[ f(x) = x^3 - 4x^2 - 7x + 28 \][/tex]
Comparing this with the provided options:
- [tex]\(f(x)=x^3 - 4 x^2 - 7 x + 28\)[/tex]
So, the correct option is:
[tex]\[ f(x)=x^3 - 4x^2 - 7x + 28 \][/tex]
Therefore, the correct answer is:
[tex]\[ f(x)=x^3 - 4 x^2 - 7 x + 28 \][/tex]
The Factor Theorem states that if [tex]\(c\)[/tex] is a zero of a polynomial [tex]\(f(x)\)[/tex], then [tex]\((x - c)\)[/tex] is a factor of [tex]\(f(x)\)[/tex].
Given the zeros are [tex]\(4\)[/tex], [tex]\(\sqrt{7}\)[/tex], and [tex]\(-\sqrt{7}\)[/tex], the polynomial can be expressed as:
[tex]\[ f(x) = (x - 4)(x - \sqrt{7})(x + \sqrt{7}) \][/tex]
First, let's simplify the factors involving the square roots:
[tex]\[ (x - \sqrt{7})(x + \sqrt{7}) = x^2 - (\sqrt{7})^2 = x^2 - 7 \][/tex]
So, the polynomial becomes:
[tex]\[ f(x) = (x - 4)(x^2 - 7) \][/tex]
Next, we need to expand this product:
[tex]\[
\begin{align*}
f(x) & = (x - 4)(x^2 - 7) \\
& = x \cdot (x^2 - 7) - 4 \cdot (x^2 - 7) \\
& = x^3 - 7x - 4x^2 + 28 \\
& = x^3 - 4x^2 - 7x + 28
\end{align*}
\][/tex]
Therefore, the polynomial function [tex]\(f(x)\)[/tex] is:
[tex]\[ f(x) = x^3 - 4x^2 - 7x + 28 \][/tex]
Comparing this with the provided options:
- [tex]\(f(x)=x^3 - 4 x^2 - 7 x + 28\)[/tex]
So, the correct option is:
[tex]\[ f(x)=x^3 - 4x^2 - 7x + 28 \][/tex]
Therefore, the correct answer is:
[tex]\[ f(x)=x^3 - 4 x^2 - 7 x + 28 \][/tex]
