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Answer :
Sure! Let's find the polynomial function with the specified roots and leading coefficient.
We need to construct a polynomial function with the roots [tex]\(\sqrt{3}\)[/tex], [tex]\(-4\)[/tex], and [tex]\(4\)[/tex] where the leading coefficient is 1.
Here’s how we can do it step-by-step:
1. Identify the Factors:
- If [tex]\(\sqrt{3}\)[/tex] is a root, then [tex]\((x - \sqrt{3})\)[/tex] is a factor.
- If [tex]\(-4\)[/tex] is a root, then [tex]\((x + 4)\)[/tex] is a factor.
- If [tex]\(4\)[/tex] is a root, then [tex]\((x - 4)\)[/tex] is a factor.
2. Multiply the Factors:
- First, multiply the factors for [tex]\(-4\)[/tex] and [tex]\(4\)[/tex]:
[tex]\((x + 4)(x - 4) = x^2 - 16\)[/tex].
This is the difference of squares.
3. Multiply with the Remaining Factor:
- Now, multiply [tex]\((x^2 - 16)\)[/tex] by [tex]\((x - \sqrt{3})\)[/tex]:
[tex]\[
(x^2 - 16)(x - \sqrt{3}) = x^3 - \sqrt{3}x^2 - 16x + 16\sqrt{3}
\][/tex]
4. Check the Given Choices:
- Compare our expanded polynomial with the given choices to find a match:
- [tex]\(f(x) = x^3 - 3x^2 - 16x + 48\)[/tex]
Therefore, the polynomial function of lowest degree with a leading coefficient of 1 and the given roots is:
[tex]\[
f(x) = x^3 - 3x^2 - 16x + 48
\][/tex]
We need to construct a polynomial function with the roots [tex]\(\sqrt{3}\)[/tex], [tex]\(-4\)[/tex], and [tex]\(4\)[/tex] where the leading coefficient is 1.
Here’s how we can do it step-by-step:
1. Identify the Factors:
- If [tex]\(\sqrt{3}\)[/tex] is a root, then [tex]\((x - \sqrt{3})\)[/tex] is a factor.
- If [tex]\(-4\)[/tex] is a root, then [tex]\((x + 4)\)[/tex] is a factor.
- If [tex]\(4\)[/tex] is a root, then [tex]\((x - 4)\)[/tex] is a factor.
2. Multiply the Factors:
- First, multiply the factors for [tex]\(-4\)[/tex] and [tex]\(4\)[/tex]:
[tex]\((x + 4)(x - 4) = x^2 - 16\)[/tex].
This is the difference of squares.
3. Multiply with the Remaining Factor:
- Now, multiply [tex]\((x^2 - 16)\)[/tex] by [tex]\((x - \sqrt{3})\)[/tex]:
[tex]\[
(x^2 - 16)(x - \sqrt{3}) = x^3 - \sqrt{3}x^2 - 16x + 16\sqrt{3}
\][/tex]
4. Check the Given Choices:
- Compare our expanded polynomial with the given choices to find a match:
- [tex]\(f(x) = x^3 - 3x^2 - 16x + 48\)[/tex]
Therefore, the polynomial function of lowest degree with a leading coefficient of 1 and the given roots is:
[tex]\[
f(x) = x^3 - 3x^2 - 16x + 48
\][/tex]
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