Only like terms combine
Two terms are like terms when they carry the same variable raised to the same power. So 4x^2 and −9x^2 are like terms; 4x^2 and 4x are not. Combining like terms just adds the coefficients and keeps the shared variable part: 4x^2 − 9x^2 = −5x^2. You never add the exponents — x^2 stays x^2.
Adding two polynomials means dropping the parentheses and gathering like terms. Because addition just merges the two lists, the parentheses come off without any sign change.
(3x^2 − 7x + 5) + (x^2 + 4x − 2) = 3x^2 + x^2 − 7x + 4x + 5 − 2 group like terms = (3+1)x^2 + (−7+4)x + (5−2) = 4x^2 − 3x + 3
Subtraction: flip every sign
Subtraction is where most slips happen. A minus sign in front of a parenthesis means subtract the whole thing, so by the distributive property it flips the sign of every term inside. Removing the parentheses of −(x^2 + 4x − 2) gives −x^2 − 4x + 2. Change all three signs — not just the first.
(3x^2 − 7x + 5) − (x^2 + 4x − 2) = 3x^2 − 7x + 5 − x^2 − 4x + 2 distribute the minus = (3−1)x^2 + (−7−4)x + (5+2) = 2x^2 − 11x + 7
The column method
For longer polynomials, stack them like numbers — each power in its own column, both written in standard form. Leave a gap where a power is missing so nothing drifts into the wrong column, then add or subtract straight down.
2x^3 + 0x^2 − 5x + 1
+ 4x^3 − 3x^2 + 2x − 6
----------------------
6x^3 − 3x^2 − 3x − 5