2x^4 -5x^3 -2x^2 -x+2 byx^2 -3x +2 without actual division

Question

prove this question without actual division.

Answer 1

Given divident is 2x^4-5x^3-2x^2-x+2

Divider is x^2-3x+2

Now factorize x^2-3x+2

x^2-2x-x+2

x(x-2)-1(x-2)

(x-2)(x-1)

Rational zeros of x^2-3x+2 is 1,2.

By synthetic division

Quotient is 2x^2+x-3 and remainder is -12x-4