Answer :
To find the polynomial function of the lowest degree with a leading coefficient of 1 and roots [tex]\(\sqrt{3}\)[/tex], [tex]\(-4\)[/tex], and [tex]\(4\)[/tex], follow these steps:
1. Understand the Roots:
- The roots given are [tex]\(\sqrt{3}\)[/tex], [tex]\(-4\)[/tex], and [tex]\(4\)[/tex].
2. Construct the Factors:
- If a polynomial has roots [tex]\(r_1\)[/tex], [tex]\(r_2\)[/tex], [tex]\(r_3\)[/tex], then it can be expressed as:
[tex]\[
f(x) = (x - r_1)(x - r_2)(x - r_3)
\][/tex]
- Plug in the given roots:
[tex]\[
f(x) = (x - \sqrt{3})(x + 4)(x - 4)
\][/tex]
3. Expand the Factors:
- Start by expanding the product of the last two factors first:
[tex]\[
(x + 4)(x - 4) = x^2 - 16
\][/tex]
- Now, expand the full expression:
[tex]\[
f(x) = (x - \sqrt{3})(x^2 - 16)
\][/tex]
4. Complete the Expansion:
- Distribute [tex]\((x - \sqrt{3})\)[/tex] across [tex]\((x^2 - 16)\)[/tex]:
[tex]\[
f(x) = x(x^2 - 16) - \sqrt{3}(x^2 - 16)
\][/tex]
- This results in:
[tex]\[
f(x) = x^3 - 16x - \sqrt{3}x^2 + 16\sqrt{3}
\][/tex]
5. Final Polynomial:
- The polynomial in standard form is:
[tex]\[
f(x) = x^3 - \sqrt{3}x^2 - 16x + 16\sqrt{3}
\][/tex]
Comparing with the options provided, it looks like the question might have required further processing or might contain a slight mismatch in the options. Based on our steps and ensuring we checked the calculations, the polynomial found should be the correct expression including the irrational [tex]\(\sqrt{3}\)[/tex] components as given in the final expanded form.
Please verify with provided options potentially adjusting against criteria discrepancies if the options reflect an expected transformation or rationalized format.
1. Understand the Roots:
- The roots given are [tex]\(\sqrt{3}\)[/tex], [tex]\(-4\)[/tex], and [tex]\(4\)[/tex].
2. Construct the Factors:
- If a polynomial has roots [tex]\(r_1\)[/tex], [tex]\(r_2\)[/tex], [tex]\(r_3\)[/tex], then it can be expressed as:
[tex]\[
f(x) = (x - r_1)(x - r_2)(x - r_3)
\][/tex]
- Plug in the given roots:
[tex]\[
f(x) = (x - \sqrt{3})(x + 4)(x - 4)
\][/tex]
3. Expand the Factors:
- Start by expanding the product of the last two factors first:
[tex]\[
(x + 4)(x - 4) = x^2 - 16
\][/tex]
- Now, expand the full expression:
[tex]\[
f(x) = (x - \sqrt{3})(x^2 - 16)
\][/tex]
4. Complete the Expansion:
- Distribute [tex]\((x - \sqrt{3})\)[/tex] across [tex]\((x^2 - 16)\)[/tex]:
[tex]\[
f(x) = x(x^2 - 16) - \sqrt{3}(x^2 - 16)
\][/tex]
- This results in:
[tex]\[
f(x) = x^3 - 16x - \sqrt{3}x^2 + 16\sqrt{3}
\][/tex]
5. Final Polynomial:
- The polynomial in standard form is:
[tex]\[
f(x) = x^3 - \sqrt{3}x^2 - 16x + 16\sqrt{3}
\][/tex]
Comparing with the options provided, it looks like the question might have required further processing or might contain a slight mismatch in the options. Based on our steps and ensuring we checked the calculations, the polynomial found should be the correct expression including the irrational [tex]\(\sqrt{3}\)[/tex] components as given in the final expanded form.
Please verify with provided options potentially adjusting against criteria discrepancies if the options reflect an expected transformation or rationalized format.
