When p (x) = x^3+ax+b is divided by x-1 the remainder is 2. When px is divided by x-2 the remainder is 1. Find the values of a and b?

a = - 8 and b = 9

Using the Remainder Theorem.

p (x) divided by (x - a) has remainder p (a)

p (1) = 1³ + a + b = 2⇒ 1 + a + b = 2 ⇒ a + b = 1 ... (1)

p (2) = 2³ + 2a + b = 1 ⇒ 8 + 2a + b = 1 ⇒ 2a + b = - 7 ... (2)

(2) - (1) gives a = - 8

substitute a = - 8 into (1) : - 8 + b = 1 ⇒ b=9