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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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