Question

If x-1 is a factor of 2x^3+3x^2+kx+3, then what is k?

Answer 1

(x-1)(a x^2 + b x + c) = 2 x^3 + 3 x^2 + k x + 3

a x^3 + bx^2 + c x - ax^2 - bx - c = 2 x^3 + 3 x^2 + k x + 3
c x - b x = k x
k = c - b
but c = -3
so
a x^3 + bx^2 - 3 x - ax^2 - bx + 3 = 2 x^3 + 3 x^2 + k x + 3
but a = 2
so
2 x^3 + bx^2 - 3 x - 2x^2 - bx + 3 = 2 x^3 + 3 x^2 + k x + 3
a x^3 + bx^2 + c x - ax^2 - bx - c = 2 x^3 + 3 x^2 + k x + 3
so b-a = 3
b-2 = 3
b = 5
k = -3 -5 = -8