Answer :
Certainly! To determine which polynomial function has the zeros [tex]\(4\)[/tex], [tex]\(\sqrt{7}\)[/tex], and [tex]\(-\sqrt{7}\)[/tex] using the Factor Theorem, follow these steps:
1. Understand the Factor Theorem:
- 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 that polynomial.
2. Find the polynomial form:
- Given the zeros [tex]\(4\)[/tex], [tex]\(\sqrt{7}\)[/tex], and [tex]\(-\sqrt{7}\)[/tex], we can express these as factors of the polynomial:
[tex]\[
(x - 4), (x - \sqrt{7}), \text{and } (x + \sqrt{7})
\][/tex]
- Hence, the polynomial can be written as:
[tex]\[
f(x) = (x - 4)(x - \sqrt{7})(x + \sqrt{7})
\][/tex]
3. Multiply these factors to form the polynomial:
- First, multiply the conjugate pairs:
[tex]\[
(x - \sqrt{7})(x + \sqrt{7}) = x^2 - (\sqrt{7})^2 = x^2 - 7
\][/tex]
- Now, multiply this result by the remaining factor:
[tex]\[
(x^2 - 7)(x - 4)
\][/tex]
- Distribute to expand the expression:
[tex]\[
x^2(x - 4) - 7(x - 4) = x^3 - 4x^2 - 7x + 28
\][/tex]
4. Compare the given polynomials:
- The expanded polynomial is:
[tex]\[
x^3 - 4x^2 - 7x + 28
\][/tex]
- Let's compare this result with the provided options:
- [tex]\( f(x) = x^3 - 4x^2 + 7x + 28 \)[/tex]
- [tex]\( f(x) = x^3 - 4x^2 - 7x + 28 \)[/tex]
- [tex]\( f(x) = x^3 + 4x^2 - 7x + 28 \)[/tex]
- [tex]\( f(x) = x^3 + 4x^2 - 7x - 28 \)[/tex]
5. Identify the correct polynomial:
- The polynomial [tex]\( f(x) = x^3 - 4x^2 - 7x + 28 \)[/tex] matches the expanded form derived from the given zeros.
Therefore, the correct polynomial function with the zeros [tex]\(4\)[/tex], [tex]\(\sqrt{7}\)[/tex], and [tex]\(-\sqrt{7}\)[/tex] is [tex]\(\boxed{x^3 - 4x^2 - 7x + 28}\)[/tex].
1. Understand the Factor Theorem:
- 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 that polynomial.
2. Find the polynomial form:
- Given the zeros [tex]\(4\)[/tex], [tex]\(\sqrt{7}\)[/tex], and [tex]\(-\sqrt{7}\)[/tex], we can express these as factors of the polynomial:
[tex]\[
(x - 4), (x - \sqrt{7}), \text{and } (x + \sqrt{7})
\][/tex]
- Hence, the polynomial can be written as:
[tex]\[
f(x) = (x - 4)(x - \sqrt{7})(x + \sqrt{7})
\][/tex]
3. Multiply these factors to form the polynomial:
- First, multiply the conjugate pairs:
[tex]\[
(x - \sqrt{7})(x + \sqrt{7}) = x^2 - (\sqrt{7})^2 = x^2 - 7
\][/tex]
- Now, multiply this result by the remaining factor:
[tex]\[
(x^2 - 7)(x - 4)
\][/tex]
- Distribute to expand the expression:
[tex]\[
x^2(x - 4) - 7(x - 4) = x^3 - 4x^2 - 7x + 28
\][/tex]
4. Compare the given polynomials:
- The expanded polynomial is:
[tex]\[
x^3 - 4x^2 - 7x + 28
\][/tex]
- Let's compare this result with the provided options:
- [tex]\( f(x) = x^3 - 4x^2 + 7x + 28 \)[/tex]
- [tex]\( f(x) = x^3 - 4x^2 - 7x + 28 \)[/tex]
- [tex]\( f(x) = x^3 + 4x^2 - 7x + 28 \)[/tex]
- [tex]\( f(x) = x^3 + 4x^2 - 7x - 28 \)[/tex]
5. Identify the correct polynomial:
- The polynomial [tex]\( f(x) = x^3 - 4x^2 - 7x + 28 \)[/tex] matches the expanded form derived from the given zeros.
Therefore, the correct polynomial function with the zeros [tex]\(4\)[/tex], [tex]\(\sqrt{7}\)[/tex], and [tex]\(-\sqrt{7}\)[/tex] is [tex]\(\boxed{x^3 - 4x^2 - 7x + 28}\)[/tex].
