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Answer :
To find the polynomial function of the lowest degree with a leading coefficient of 1 and roots [tex]\( \sqrt{3}, -4 \)[/tex], and [tex]\( 4 \)[/tex], we proceed as follows:
1. Identify the Roots: The roots of the polynomial are [tex]\( \sqrt{3} \)[/tex], [tex]\(-4\)[/tex], and [tex]\( 4 \)[/tex].
2. Form the Factors: For each root [tex]\( r \)[/tex], the factor of the polynomial is [tex]\( (x - r) \)[/tex]. Thus, for the given roots, the factors will be:
- [tex]\( (x - \sqrt{3}) \)[/tex]
- [tex]\( (x + 4) \)[/tex]
- [tex]\( (x - 4) \)[/tex]
3. Multiply the Factors: The polynomial is formed by multiplying all these factors together:
[tex]\[
(x - \sqrt{3})(x + 4)(x - 4)
\][/tex]
4. Expand the Product:
First, multiply [tex]\( (x + 4)(x - 4) \)[/tex] using the difference of squares:
[tex]\[
(x + 4)(x - 4) = x^2 - 16
\][/tex]
Next, multiply this result by the remaining factor:
[tex]\[
(x - \sqrt{3})(x^2 - 16)
\][/tex]
Distribute [tex]\( (x - \sqrt{3}) \)[/tex] across [tex]\( (x^2 - 16) \)[/tex]:
[tex]\[
x(x^2 - 16) - \sqrt{3}(x^2 - 16)
\][/tex]
Perform the distribution:
[tex]\[
x^3 - 16x - \sqrt{3}x^2 + 16\sqrt{3}
\][/tex]
5. Combine Like Terms: Collect the terms together to write the polynomial:
- The [tex]\( x^3 \)[/tex] term is [tex]\( x^3 \)[/tex]
- The [tex]\( x^2 \)[/tex] term is [tex]\(-\sqrt{3}x^2\)[/tex]
- The [tex]\( x \)[/tex] term is [tex]\(-16x\)[/tex]
- The constant term is [tex]\( 16\sqrt{3} \)[/tex]
So the polynomial is:
[tex]\[
x^3 - \sqrt{3}x^2 - 16x + 16\sqrt{3}
\][/tex]
6. Verify Leading Coefficient: Check that the leading coefficient is 1, which it is in this case.
Therefore, the polynomial function that satisfies the given conditions is:
[tex]\[
f(x) = x^3 - \sqrt{3}x^2 - 16x + 16\sqrt{3}
\][/tex]
This expression matches our derived polynomial and represents the function with the specified roots and leading coefficient of 1.
1. Identify the Roots: The roots of the polynomial are [tex]\( \sqrt{3} \)[/tex], [tex]\(-4\)[/tex], and [tex]\( 4 \)[/tex].
2. Form the Factors: For each root [tex]\( r \)[/tex], the factor of the polynomial is [tex]\( (x - r) \)[/tex]. Thus, for the given roots, the factors will be:
- [tex]\( (x - \sqrt{3}) \)[/tex]
- [tex]\( (x + 4) \)[/tex]
- [tex]\( (x - 4) \)[/tex]
3. Multiply the Factors: The polynomial is formed by multiplying all these factors together:
[tex]\[
(x - \sqrt{3})(x + 4)(x - 4)
\][/tex]
4. Expand the Product:
First, multiply [tex]\( (x + 4)(x - 4) \)[/tex] using the difference of squares:
[tex]\[
(x + 4)(x - 4) = x^2 - 16
\][/tex]
Next, multiply this result by the remaining factor:
[tex]\[
(x - \sqrt{3})(x^2 - 16)
\][/tex]
Distribute [tex]\( (x - \sqrt{3}) \)[/tex] across [tex]\( (x^2 - 16) \)[/tex]:
[tex]\[
x(x^2 - 16) - \sqrt{3}(x^2 - 16)
\][/tex]
Perform the distribution:
[tex]\[
x^3 - 16x - \sqrt{3}x^2 + 16\sqrt{3}
\][/tex]
5. Combine Like Terms: Collect the terms together to write the polynomial:
- The [tex]\( x^3 \)[/tex] term is [tex]\( x^3 \)[/tex]
- The [tex]\( x^2 \)[/tex] term is [tex]\(-\sqrt{3}x^2\)[/tex]
- The [tex]\( x \)[/tex] term is [tex]\(-16x\)[/tex]
- The constant term is [tex]\( 16\sqrt{3} \)[/tex]
So the polynomial is:
[tex]\[
x^3 - \sqrt{3}x^2 - 16x + 16\sqrt{3}
\][/tex]
6. Verify Leading Coefficient: Check that the leading coefficient is 1, which it is in this case.
Therefore, the polynomial function that satisfies the given conditions is:
[tex]\[
f(x) = x^3 - \sqrt{3}x^2 - 16x + 16\sqrt{3}
\][/tex]
This expression matches our derived polynomial and represents the function with the specified roots and leading coefficient of 1.
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