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------------------------------------------------ What is the polynomial function of the lowest degree with a leading coefficient of 1 and roots [tex]\sqrt{3}[/tex], -4, and 4?

A. [tex]f(x) = x^3 - 3x^2 + 16x + 48[/tex]
B. [tex]f(x) = x^3 - 3x^2 - 16x + 48[/tex]
C. [tex]f(x) = x^4 - 19x^2 + 48[/tex]
D. [tex]f(x) = x^4 - 13x^2 + 48[/tex]

Answer :

Final answer:

The polynomial function with the given roots sqrt(3), -4, and 4 is [tex]f(x) = x^3 - 16x - \sqrt{3} x^2 + 16\sqrt{3}[/tex]

Explanation:

To find the polynomial function with the given roots, we can use the fact that if a number is a root of a polynomial function, then (x - root) is a factor of the polynomial.

So, the polynomial with roots [tex]\sqrt{3}[/tex], -4, and 4 can be written as:

[tex]f(x) = (x - \sqrt{3} )(x + 4)(x - 4)[/tex]

Expanding this expression, we get:

[tex]f(x) = (x - \sqrt{3} )(x^2 - 16)[/tex]

Finally, multiplying the leading coefficient 1, we have:

[tex]f(x) = x(x^2 - 16) - \sqrt{3} (x^2 - 16)[/tex]

Expanding further, we get:

[tex]f(x) = x^3 - 16x - \sqrt{3} x^2 + 16\sqrt{3}[/tex]

Learn more about Polynomial functions here:

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