Question

What must be subtracted from the polynomial f(x) = x⁴ + 2x³ − 13x² − 12x + 21 so that the resulting polynomial is exactly divisible by x² − 4x + 3 ?

Answer 1

We know that Dividend = Quotient x Divisor + Remainder.

Dividend - Remainder = Quotient x Divisor.

Clearly, Right hand side of the above result is divisible by the divisor.

Therefore, left hand side is also divisible by the divisor.

Thus, if we subtract remainder from the dividend, then it will be exactly divisible by the divisor.

Dividing x⁴ + 2x³ − 13x² − 12x + 21 by x² − 4x + 3

Therefore, quotient = x² + 6x + 8 and remainder = (2x - 3).

Thus, if we subtract the remainder 2x - 3 from x⁴ + 2x³ − 13x² − 12x + 21 it will be divisible by x² − 4x + 3.