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Answer :
To find the polynomial function of the lowest degree with a leading coefficient of 1 and given roots [tex]\(\sqrt{3}, -4,\)[/tex] and [tex]\(4\)[/tex], we can follow these steps:
1. Understand the Form: The polynomial should have the smallest degree possible to include all the provided roots. Since there are three unique roots, the polynomial will at least be a cubic (degree 3).
2. Write the Factor Form: Each root [tex]\((p)\)[/tex] of the polynomial corresponds to a factor of [tex]\((x - p)\)[/tex]. Using the given roots, the polynomial can be expressed as:
[tex]\[
(x - \sqrt{3})(x + 4)(x - 4)
\][/tex]
3. Multiply the Factors:
- Start by multiplying the last two linear factors, [tex]\((x + 4)\)[/tex] and [tex]\((x - 4)\)[/tex]. This is a difference of squares:
[tex]\[
(x + 4)(x - 4) = x^2 - 16
\][/tex]
- Now, multiply this result by the remaining factor, [tex]\((x - \sqrt{3})\)[/tex]:
[tex]\[
(x - \sqrt{3})(x^2 - 16)
\][/tex]
4. Expand the Polynomial:
- Distribute [tex]\((x - \sqrt{3})\)[/tex] across [tex]\(x^2 - 16\)[/tex]:
[tex]\[
x(x^2 - 16) - \sqrt{3}(x^2 - 16)
\][/tex]
- This results in:
[tex]\[
x^3 - 16x - \sqrt{3}x^2 + 16\sqrt{3}
\][/tex]
5. Simplify the Expression: Combine like terms to write the polynomial in standard form:
[tex]\[
x^3 - \sqrt{3}x^2 - 16x + 16\sqrt{3}
\][/tex]
So, the polynomial function is represented as:
[tex]\[
f(x) = x^3 - 1.73205080756888x^2 - 16.0x + 27.712812921102
\][/tex]
This matches the numerical answer we have, confirming that the polynomial we have determined is indeed the correct one based on the given roots. None of the provided choices exactly matches this form, so that might reflect an error in the option list.
1. Understand the Form: The polynomial should have the smallest degree possible to include all the provided roots. Since there are three unique roots, the polynomial will at least be a cubic (degree 3).
2. Write the Factor Form: Each root [tex]\((p)\)[/tex] of the polynomial corresponds to a factor of [tex]\((x - p)\)[/tex]. Using the given roots, the polynomial can be expressed as:
[tex]\[
(x - \sqrt{3})(x + 4)(x - 4)
\][/tex]
3. Multiply the Factors:
- Start by multiplying the last two linear factors, [tex]\((x + 4)\)[/tex] and [tex]\((x - 4)\)[/tex]. This is a difference of squares:
[tex]\[
(x + 4)(x - 4) = x^2 - 16
\][/tex]
- Now, multiply this result by the remaining factor, [tex]\((x - \sqrt{3})\)[/tex]:
[tex]\[
(x - \sqrt{3})(x^2 - 16)
\][/tex]
4. Expand the Polynomial:
- Distribute [tex]\((x - \sqrt{3})\)[/tex] across [tex]\(x^2 - 16\)[/tex]:
[tex]\[
x(x^2 - 16) - \sqrt{3}(x^2 - 16)
\][/tex]
- This results in:
[tex]\[
x^3 - 16x - \sqrt{3}x^2 + 16\sqrt{3}
\][/tex]
5. Simplify the Expression: Combine like terms to write the polynomial in standard form:
[tex]\[
x^3 - \sqrt{3}x^2 - 16x + 16\sqrt{3}
\][/tex]
So, the polynomial function is represented as:
[tex]\[
f(x) = x^3 - 1.73205080756888x^2 - 16.0x + 27.712812921102
\][/tex]
This matches the numerical answer we have, confirming that the polynomial we have determined is indeed the correct one based on the given roots. None of the provided choices exactly matches this form, so that might reflect an error in the option list.
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