If the polynomials ax 3 -7x 2 +7x-2 and x 3 -2ax 2 +8x-8 leave the same remainder when divided by x-2. Then the value of a is.

By the Remainder Theorem, the remainder when a polynomial f(x) is divided by (x - c) is f(c). Here c = 2, so we only need to evaluate both polynomials at x = 2 and equate the results.

For the first polynomial, P(2) = a(8) - 7(4) + 7(2) - 2 = 8a - 28 + 14 - 2 = 8a - 16.

For the second, Q(2) = 8 - 2a(4) + 8(2) - 8 = 8 - 8a + 16 - 8 = 16 - 8a.

Setting P(2) = Q(2): 8a - 16 = 16 - 8a, so 16a = 32, giving a = 2. Option C is correct.

• A) 0: gives remainders -16 and 16, which are not equal.
• B) 1: gives remainders -8 and 8, which are not equal.
• D) -1: gives remainders -24 and 24, which are not equal.

The core concept is the Remainder Theorem, a basic algebra tool that also underlies polynomial factorisation and synthetic division. It is a quick method for Prelims, and the arithmetic is where most marks are lost.

Trap: Students often start a full long division or try to find quotients. Simply substitute x = 2 in both polynomials, and take care with the sign of the -8a term in the second polynomial.