Answer:
C
Step-by-step 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
Answer:
x^3 + x^2 - 26x + 24.
Step-by-step 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).
