Question
The polynomials ax3 - 3x2 + 4 and 2x3 - 5x + a when divided by (x - 2) leave the remainders p and q respectively. If p - 2q = 4, find the value of a.
Solution
Let f(x) = ax3 - 3x2 + 4
and g(x) = 2x3 - 5x + a
By remainder theorem,
f(2) = p ...(1) | x - 2 = 0 ⇒ x = 2
and g(2) = q ...(2) | x - 2 = 0 ⇒ x = 2
(1) gives
a(2)3 - 3(2)2 + 4 = p
⇒ 8a - 8 = p ...(3)
(2) gives
2(2)3 - 5(2) + a = q
⇒ 6 + a = q ...(4)
According to the question,
p - 2q = 4
⇒ (8a - 8) - 2 (6 + a) = 4
⇒ 8a - 8 - 12 - 2a = 4
⇒ 6a = 24
⇒ a = 4
