College

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

A. \( f(x) = (x - \sqrt{3})(x + 4)(x - 4) \)

B. \( f(x) = x^3 - 3x^2 + 16x + 48 \)

C. \( f(x) = x^3 - 3x^2 - 16x + 48 \)

D. \( f(x) = x^3 - 19x^2 + 48 \)

E. \( f(x) = x^3 - 13x^2 + 48 \)

Answer :

Final answer:

The polynomial function of lowest degree with a leading coefficient of 1 and given roots is x^4 - 18x^2 + 32.


Explanation:

The polynomial function of lowest degree with a leading coefficient of 1 and roots √√3, -4, and 4 can be found by using the fact that if a number is a root of a polynomial, then (x - root) is a factor of the polynomial. Therefore, the polynomial can be written as (x - √√3)(x + 4)(x - 4) = (x^2 - 2√3)(x^2 - 16) = x^4 - 18x^2 + 32.


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