Let f(x) = 2x3 + ax2 + 4x – 12 and g(x) = x3 + x2 – 2x + a
When f(x) is divided by x – 3, the remainder is f(3).
Now f(3) = 2(3)3 + a(3)2 + 4(3) – 12 = 54 + 9a + 12 – 12
f(3) = 9a + 54 … (1)
When g(x) is divided by x – 3, the remainder is g(3).
Now g(3) = 33 + 32 – 2(3) + a = 27 + 9 – 6 + a
g(3) = a + 30 … (2)
Since, the remainder’s are same (1) = (2)
Given that f(3) = g(3)
That is 9a + 54 = a + 30
9a – a = 30 – 54 ⇒ 8a = -24 ∴ a = -3
Substituting a = -3 in f(3), we get
f(3) = 9(-3) + 54 = -27 + 54
f(3) = 27
∴ The remainder is 27