Question

Solve x^3 – 4x^2 + x + 6 = 0. Show all work.

x = –1, –2, –3

x = –1, 2, 3

x = 1, 2, 3

x = 1, –2, –3

Answer 1

We look for rational roots among ±1, ±2, ±3, ±6. Test x = −1:
(−1)^3 − 4(−1)^2 + (−1) + 6 = −1 − 4 − 1 + 6 = 0,
so x = −1 is a root.

Divide the cubic by (x + 1) (synthetic division with −1):

Coefficients: 1 , −4 , 1 , 6
Bring down 1.
−1·1 = −1 → add to −4 gives −5.
−1·(−5) = 5 → add to 1 gives 6.
−1·6 = −6 → add to 6 gives 0.

Quotient is x^2 − 5x + 6. Factor:
x^2 − 5x + 6 = (x − 2)(x − 3).

Thus the roots are x = −1, 2, 3.

Answer: x = −1, 2, 3.