Question

Divide x^3 + 3x^2y + 4xy^2 + 2y^3 by x + y.

x^2 – y^2

x^2 + 2xy + 2y^2

x^2 + 2xy + 2y^2 R. 4y^3

Answer 1

Divide P(x,y) = x^3 + 3x^2y + 4xy^2 + 2y^3 by (x + y).

Let quotient be x^2 + Axy + By^2. Then
(x+y)(x^2 + Axy + By^2) = x^3 + (A+1)x^2y + (B+A)xy^2 + By^3.

Match coefficients to P:
A+1 = 3 → A = 2
B+A = 4 → B + 2 = 4 → B = 2
B = 2 matches the y^3 term.

So quotient = x^2 + 2xy + 2y^2 and remainder = 0.

Answer: x^2 + 2xy + 2y^2 (remainder 0).