If A = x2 (1 - 3y2), B = x (xy2 - 2x) and C = 3y (y - 4x2y), then A + B - C =…

Concept

To compare two polynomial expressions written in the same variables, expand every product with the distributive law, then group like terms (same variable part and exponents) and add their numerical coefficients. Two polynomials are equal only when each like term's coefficient matches, so the coefficient of x2 equals p, the coefficient of x2 y2 equals q, and the coefficient of y2 equals r.

Application

1. Expand A = x2 (1 - 3y2) = x2 - 3x2 y2.

2. Expand B = x (xy2 - 2x) = x2 y2 - 2x2.

3. Expand C = 3y (y - 4x2 y) = 3y2 - 12x2 y2.

4. Form A + B - C = (x2 - 3x2 y2) + (x2 y2 - 2x2) - (3y2 - 12x2 y2); subtracting C flips the sign of each of its terms.

5. Group like terms: x2 terms give x2 - 2x2 = -x2; x2 y2 terms give -3x2 y2 + x2 y2 + 12x2 y2 = 10x2 y2; y2 terms give -3y2.

6. So A + B - C = -x2 + 10x2 y2 - 3y2, and matching with p x2 + q x2 y2 + r y2 gives p = -1, q = 10, r = -3.

7. Therefore 5p + q - r = 5(-1) + 10 - (-3) = -5 + 10 + 3 = 8.

Cross-check

Cross-check by substituting x = 1, y = 1: A = 1(1-3) = -2, B = 1(1-2) = -1, C = 3(1-4) = -9, so A + B - C = -2 - 1 + 9 = 6. The expression -x2 + 10x2 y2 - 3y2 at x = y = 1 also gives -1 + 10 - 3 = 6, confirming the coefficients.