Overview
A polynomial groups terms by powers of its variables. Addition combines matching powers, and subtraction first reverses every sign in the polynomial being removed. Coefficients can be negative or zero, but different powers remain separate terms.
Consider this example: Add (2x² − 3x + 4) and (x² + 5x − 7). Quadratic terms give 3x², linear terms give 2x, and constants give −3. Thus the sum is 3x² + 2x − 3. The linear coefficient 8 ignores the sign on −3x.
The method also handles a different case: Subtract (2x² − x − 5) from (5x² + 3x − 2). Write 5x² + 3x − 2 − 2x² + x + 5. Combining yields 3x² + 4x + 3. The result with constant −7 fails to reverse the subtracted −5.
For a further application, consider: Let P(x) = ax² + bx + c and Q(x) = 2x² + 3x + 5. If P(x) − Q(x) has no quadratic term and no constant term, and P(2) = 15, what is b? The missing quadratic term means a = 2; the missing constant means c = 5. Then P(2) = 4a + 2b + c = 8 + 2b + 5 = 15, so b = 1. The linear coefficient 3 in Q need not cancel because the difference is allowed to have a linear term.
Combining x² with x treats unlike factors as the same quantity. Losing the subtraction on a negative constant creates the wrong final constant. Align equal degrees before combining. In a new subtraction, rewrite the removed polynomial with all signs reversed, then test the result at zero to check its constant term.