The remainder, when x 3 -6x 2 +2x-4 is divided by -3/2x+1 is

Take the divisor and set it to zero: -3/2 x + 1 = 0, so x = 2/3. By the Remainder Theorem, the remainder is the polynomial's value at this root, p(2/3). Here p(x) = x^3 - 6x^2 + 2x - 4.

p(2/3) = 8/27 - 6(4/9) + 4/3 - 4 = 8/27 - 72/27 + 36/27 - 108/27 = -136/27.

Why the others fail:

• A (136/27): Only the sign is flipped. Careless sign handling on the -6x^2 and -4 terms gives this.
• C (136): The denominator 27 is dropped and the sign is wrong. Fractions were not carried through.
• D (-136): The correct numerator with the denominator lost. Working in decimals or rounding can produce this.

The concept is the Remainder Theorem, which applies to any divisor of the form (ax + b) through the root of the divisor. It appears often in Prelims maths and reasoning-based algebra, and it is a quick check for Mains-level polynomial problems too.

Trap: candidates use the root x = -1 or x = -2/3, or use the coefficient 3/2 directly as if the divisor were x - 3/2. Set the divisor itself to zero and then substitute.