Question

if the polynomial ax^3 +4x^2+3x-4 and x^3-4x+a leave same remainder when divided by x-3, find a

Answer 1

Given ax3 + 4x2 + 3x - 4 and x3 - 4x + a leave the same remainder when divided by x - 3.

Let p(x) = ax3 + 4x2 + 3x - 4 and g(x) = x3 - 4x + a

By remainder theorem, if f(x) is divided by (x − a) then the remainder is f(a)

Here when p(x) and g(x) are divided by (x − 3) the remainders are p(3) and g(3) respectively.

Also given that p(3) = g(3) �→ (1)

Put x = 3 in both p(x) and g(x)

Hence equation (1) becomes,

a(3)3 + 4(3)2 + 3(3) - 4 = (3)3 - 4(3) + a

⇒ 27a + 36 + 9 − 4 = 27 − 12 + a

⇒ 27a + 41 = 15 + a

⇒ 26a �= 15 − 41 = − 26

∴ a = −1