Any division statement can be rewritten in the form of
Dividend = Divisor x Quotient + Remainder.
When the remainder is zero, the dividend is said to be the multiple of the divisor.
When the remainder is not equal to zero, and if it is subtracted from the dividend, then the dividend would become a multiple of divisor.
Dividend = x^4+2x^3-13x^2-12x+21
Divisor = x^2-4x+3
Step 1: Divide the dividend by divisor
Quotient = x^2+6x+8
Remainder = 2x – 3
Step 2: Hence, on subtracting 2x – 3 from the dividend, it will be exactly divisible by the given divisor.
Verification:
On subtracting 2x – 3 from the dividend, we get
x^4+2x^3-13x^2-12x+21-(2x-3) = x^4+2x^3-13x^2-14x+24
Dividing x^4+2x^3-13x^2-14x+24 by x^2-4x+3
Quotient = x^2+6x+8
Remainder = 0