Question

Algebraically determine the values of h and k to correctly complete the identity stated below 3x^3-6x^2+4x-2=(x-2)(3x^2+hx+4)+k

Answer 1

We want to find h and k so that
3x³ - 6x² + 4x - 2 ≡ (x-2)(3x² + hx + 4) + k

Expand the right side.
(x-2)(3x² + hx + 4) + k
= 3x³ + hx² + 4x - 6x² - 2hx - 8 + k
= 3x³ + (h-6)x² + (4 - 2h)x + k - 8

Match coefficients of x²:
h - 6 = -6
h = 0

Match coefficients of x:
4 - 2h = 4    (Correct because h=0)

Match constants.
k - 8 = -2
k = 6

Answer:  h=0, k=6