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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], you need to follow these steps:
1. Identify the Polynomial Roots:
The given roots are [tex]\(\sqrt{3}\)[/tex], [tex]\(-4\)[/tex], and [tex]\(4\)[/tex].
2. Construct Factors from the Roots:
Each root [tex]\(r\)[/tex] of the polynomial gives a factor of the form [tex]\((x - r)\)[/tex]. So, the factors based on the roots are:
- For [tex]\(\sqrt{3}\)[/tex]: [tex]\((x - \sqrt{3})\)[/tex]
- For [tex]\(-4\)[/tex]: [tex]\((x + 4)\)[/tex]
- For [tex]\(4\)[/tex]: [tex]\((x - 4)\)[/tex]
3. Form the Polynomial by Multiplying the Factors:
Multiply all these factors together to form the polynomial:
[tex]\[
f(x) = (x - \sqrt{3})(x + 4)(x - 4)
\][/tex]
4. Expand the Polynomial:
First, multiply the last two factors:
[tex]\((x + 4)(x - 4) = x^2 - 16\)[/tex]
Now, expand this product with the first factor:
[tex]\((x - \sqrt{3})(x^2 - 16)\)[/tex]
5. Simplify:
Expand the expression:
[tex]\[
= x(x^2 - 16) - \sqrt{3}(x^2 - 16)
\][/tex]
[tex]\[
= x^3 - 16x - \sqrt{3}x^2 + 16\sqrt{3}
\][/tex]
Arrange the terms:
[tex]\[
= x^3 - \sqrt{3}x^2 - 16x + 16\sqrt{3}
\][/tex]
So, the polynomial function you are looking for is:
[tex]\[
f(x) = x^3 - \sqrt{3}x^2 - 16x + 16\sqrt{3}
\][/tex]
This polynomial function matches the requirement of having a leading coefficient of 1 and includes the roots [tex]\(\sqrt{3}\)[/tex], [tex]\(-4\)[/tex], and [tex]\(4\)[/tex].
1. Identify the Polynomial Roots:
The given roots are [tex]\(\sqrt{3}\)[/tex], [tex]\(-4\)[/tex], and [tex]\(4\)[/tex].
2. Construct Factors from the Roots:
Each root [tex]\(r\)[/tex] of the polynomial gives a factor of the form [tex]\((x - r)\)[/tex]. So, the factors based on the roots are:
- For [tex]\(\sqrt{3}\)[/tex]: [tex]\((x - \sqrt{3})\)[/tex]
- For [tex]\(-4\)[/tex]: [tex]\((x + 4)\)[/tex]
- For [tex]\(4\)[/tex]: [tex]\((x - 4)\)[/tex]
3. Form the Polynomial by Multiplying the Factors:
Multiply all these factors together to form the polynomial:
[tex]\[
f(x) = (x - \sqrt{3})(x + 4)(x - 4)
\][/tex]
4. Expand the Polynomial:
First, multiply the last two factors:
[tex]\((x + 4)(x - 4) = x^2 - 16\)[/tex]
Now, expand this product with the first factor:
[tex]\((x - \sqrt{3})(x^2 - 16)\)[/tex]
5. Simplify:
Expand the expression:
[tex]\[
= x(x^2 - 16) - \sqrt{3}(x^2 - 16)
\][/tex]
[tex]\[
= x^3 - 16x - \sqrt{3}x^2 + 16\sqrt{3}
\][/tex]
Arrange the terms:
[tex]\[
= x^3 - \sqrt{3}x^2 - 16x + 16\sqrt{3}
\][/tex]
So, the polynomial function you are looking for is:
[tex]\[
f(x) = x^3 - \sqrt{3}x^2 - 16x + 16\sqrt{3}
\][/tex]
This polynomial function matches the requirement of having a leading coefficient of 1 and includes the roots [tex]\(\sqrt{3}\)[/tex], [tex]\(-4\)[/tex], and [tex]\(4\)[/tex].
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