jmc

Question

Let P(x) be a cubic polynomial such that P(0) = -3 and P(1) = 4. When P(x) is divided by x^2 + x + 1, the remainder is 2x - 1. What is the quotient when P(x) is divided by x^2 + x + 1?

Accepted Answer

Let the quotient be ax + b, so P(x) = (ax + b)(x^2 + x + 1) + 2x - 1.Setting x = 0, we get -3 = b - 1.Setting x = 1, we get 4 = 3(a + b) + 1.Then b = -2, so 4 = 3(a - 2) + 1. Solving, we find a = 3. Hence, the quotient is 3x - 2.