High School

What is the polynomial function of the lowest degree with a leading coefficient of 1 and roots √3, -4, and 4?

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

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

C) \( f(x) = x^4 - 19x^2 + 48 \)

D) \( f(x) = x^4 - 13x^2 + 48 \)

Answer :

Final answer:

The polynomial function with the lowest degree and a leading coefficient of 1, with roots √3, -4, and 4, is f(x) = (x² - 3)(x + 4)(x - 4).

Explanation:

To find the polynomial function with the lowest degree and a leading coefficient of 1, we can use the fact that the roots are √3, -4, and 4. Since square roots are involved, the degree of the polynomial will be at least 2. We know that the factors of the polynomial will be (x - √3), (x + 4), and (x - 4) because those are the values that make the polynomial equal to 0 at its roots. Multiplying these factors together gives us the polynomial function of lowest degree:

f(x) = (x - √3)(x + 4)(x - 4)

Simplifying further, we get:

f(x) = (x² - 3)(x + 4)(x - 4)