Question

Which correctly describes the roots of the following cubic equation? x^3 − 3x^2 + 4x − 12 = 0

A. Two real roots and one complex root
B. One real root and two complex roots
C. Three real roots, each with a different value
D. Three real roots, two of which are equal in value

Answer 1

x = 3 works
divide x^3 − 3x^2 + 4x − 12 by (x-3)
get
x^2 + 4
so
(x-3)(x^2+4) works
so
x = 3 or 2i or -2 i

Answer 2

One real root, x = 3
Two complex roots, 2 i and -2 i

Answer 3

D. Three real roots, two of which are equal in value

Answer 4

Apologies for the error in my previous response.

You are correct. Dividing the equation x^3 − 3x^2 + 4x − 12 = 0 by (x-3) yields x^2 + 4. This means that (x-3)(x^2 + 4) = 0. Solving for x, we have three possible solutions:
1. x-3 = 0 ---> x = 3
2. x^2 + 4 = 0 ---> x^2 = -4 ---> x = ±2i

Therefore, the correct answer is A. Two real roots (x=3) and one complex root (x=2i and x=-2i).

Answer 5

Apologies for the confusion, you are correct. The correct answer is:

B. One real root, x = 3 and two complex roots, 2i and -2i.