Answer :
To find a cubic function with the given zeros [tex]\(\sqrt{7}\)[/tex], [tex]\(-\sqrt{7}\)[/tex], and [tex]\(-4\)[/tex], we can follow these steps:
1. Understand the Zeros: We have three zeros for the cubic function: [tex]\(\sqrt{7}\)[/tex], [tex]\(-\sqrt{7}\)[/tex], and [tex]\(-4\)[/tex].
2. Form the Factors: Each zero corresponds to a factor of the function. Thus, the factors are:
- [tex]\((x - \sqrt{7})\)[/tex]
- [tex]\((x + \sqrt{7})\)[/tex]
- [tex]\((x + 4)\)[/tex]
3. Construct the Polynomial: Multiply these factors together to form the polynomial:
[tex]\[
f(x) = (x - \sqrt{7})(x + \sqrt{7})(x + 4)
\][/tex]
4. Simplify the Expression: First, multiply the factors [tex]\((x - \sqrt{7})(x + \sqrt{7})\)[/tex]. This is a difference of squares, which gives:
[tex]\[
(x - \sqrt{7})(x + \sqrt{7}) = x^2 - (\sqrt{7})^2 = x^2 - 7
\][/tex]
5. Expand the Polynomial: Now multiply the result by the remaining factor [tex]\((x + 4)\)[/tex]:
[tex]\[
f(x) = (x^2 - 7)(x + 4)
\][/tex]
Distribute to find the expanded polynomial:
[tex]\[
f(x) = x^2 \cdot x + x^2 \cdot 4 - 7 \cdot x - 7 \cdot 4
\][/tex]
[tex]\[
f(x) = x^3 + 4x^2 - 7x - 28
\][/tex]
Thus, the cubic function with the given zeros is:
[tex]\[ f(x) = x^3 + 4x^2 - 7x - 28 \][/tex]
This matches one of the options provided:
- [tex]\( f(x) = x^3 + 4x^2 - 7x - 28 \)[/tex]
1. Understand the Zeros: We have three zeros for the cubic function: [tex]\(\sqrt{7}\)[/tex], [tex]\(-\sqrt{7}\)[/tex], and [tex]\(-4\)[/tex].
2. Form the Factors: Each zero corresponds to a factor of the function. Thus, the factors are:
- [tex]\((x - \sqrt{7})\)[/tex]
- [tex]\((x + \sqrt{7})\)[/tex]
- [tex]\((x + 4)\)[/tex]
3. Construct the Polynomial: Multiply these factors together to form the polynomial:
[tex]\[
f(x) = (x - \sqrt{7})(x + \sqrt{7})(x + 4)
\][/tex]
4. Simplify the Expression: First, multiply the factors [tex]\((x - \sqrt{7})(x + \sqrt{7})\)[/tex]. This is a difference of squares, which gives:
[tex]\[
(x - \sqrt{7})(x + \sqrt{7}) = x^2 - (\sqrt{7})^2 = x^2 - 7
\][/tex]
5. Expand the Polynomial: Now multiply the result by the remaining factor [tex]\((x + 4)\)[/tex]:
[tex]\[
f(x) = (x^2 - 7)(x + 4)
\][/tex]
Distribute to find the expanded polynomial:
[tex]\[
f(x) = x^2 \cdot x + x^2 \cdot 4 - 7 \cdot x - 7 \cdot 4
\][/tex]
[tex]\[
f(x) = x^3 + 4x^2 - 7x - 28
\][/tex]
Thus, the cubic function with the given zeros is:
[tex]\[ f(x) = x^3 + 4x^2 - 7x - 28 \][/tex]
This matches one of the options provided:
- [tex]\( f(x) = x^3 + 4x^2 - 7x - 28 \)[/tex]
