Question

One root of f (x) = x cubed minus 4 x squared minus 20 x + 48 is x = 6. What are all the factors of the function? Use the Remainder Theorem.

(x + 6)(x + 8)
(x – 6)(x – 8)
(x – 2)(x + 4)(x – 6)
(x + 2)(x – 4)(x + 6)

Answer 1

Using the Remainder Theorem and long division/synthetic division, the polynomial f(x) can be factored into (x - 6)(x + 4)(x - 2), since x = 6 is one of its roots.

Step-by-step explanation:

The Remainder Theorem states that if a polynomial f(x) has a root at x = r, then (x - r) is a factor of the polynomial. Given that one root of the function f(x) = x^3 - 4x^2 - 20x + 48 is x = 6, we can conclude that (x - 6) is a factor of f(x).

To find all the factors, we divide f(x) by (x - 6). Long division or synthetic division can be used for this. Performing the division, we get a quotient of x^2 + 2x - 8, which can be further factored into (x + 4)(x - 2). Hence, the complete factorization of the polynomial is (x - 6)(x + 4)(x - 2).

Therefore, the correct answer is (x - 6)(x + 4)(x - 2), and not any of the other options presented by the student.