Question

What polynomial has roots of -6, 1, and 4 ​

Answer 1

x^3 + x^2 - 26x + 24.

Explanation:

Knowing the roots we can immediately write it in factor form as follows:

f(x) = (x + 6)(x - 1)(x - 4).

Note that when f(x) = 0 each of the factors can be zero and , for example, when x + 6 = 0 then x = -6.

We now expand the expression:

(x + 6)(x - 1)(x - 4)

= (x + 6)(x^2 - 4x - 1x + 4)

= (x + 6)(x^2 - 5x + 4)

= x(x^2 - 5x + 4) + 6(x^2 - 5x + 4)

= x^3 - 5x^2 + 4x + 6x^2 - 30x + 24 Adding like terms:

= x^3 + x^2 - 26x + 24. (Answer).

Answer 2

C

Explanation:

Given the roots of the polynomial are x = - 6, x = 1 and x = 4 the the factors are

(x + 6), (x - 1) and (x - 4)

The polynomial is the product of the factors, that is

f(x) = a(x - 1)(x - 4)(x + 6) ← a is a multiplier

let a = 1 and expand the first pair of factors

f(x) = (x² - 5x + 4)(x + 6)

= x(x² - 5x + 4) + 6(x² - 5x + 4) ← distribute both parenthesis

= x³ - 5x² + 4x + 6x² - 30x + 24 ← collect like terms

f(x) = x³ + x² - 26x + 24 → C