Question
The polynomials ax3 + 3x2 - 3 and 2x3 - 5x + a when divided by x - 4 leaves the remainders p and q respectively. Find the value of a, if 2p = q.
Solution
Let f(x) = ax3 + 3x2 - 3
and g(x) = 2x3 - 5x + a
By remainder theorem,
f(4) = p ...(1) | x - 4 = 0 ⇒ x = 4
and g (4) = q ...(2) | x - 4 = 0 ⇒ x = 4
(1) gives
a(4)3 + 3(4)2 - 3 = p
⇒ 64a + 48 - 3 = p
⇒ 64a + 45 = p ...(3)
(2) gives
2(4)3 - 5(4) + a = q
⇒ 128 - 20 + a = q
⇒ 108 + a = q ...(4)
According to the question,
2p = q
2(64a + 45) = 108 + a
128a + 90 = 108 + a
128a - a = 108 - 90
127a = 18
