Given that:
x³ + kx² - kx + 1 divide by (x - 2), has a remainder of 3
Let f(x) = x³ + kx² - kx + 1
When f(x) is divided by (x 2), the remainder is =3
This is the remainder theorem:
So,
f(2) = 2³ + k(2)² - 2k + 1 = 3
8 + 4k - 2k + 1 = 3
let's collect like terms
4k - 2k + 8 + 1 = 3
2k + 9 = 3
lets subtract 9 from both sides
2k = 3 - 9
2k = -6
divide both sides by 2
k = -6/2
k = -3