Adding polynomials is collecting like terms with more kinds of term to collect. These free printable worksheets ask for the result written out term by term — the x² part, the x part and the number — so nothing can hide inside a half-finished answer.
The x² terms combine with each other and the x terms with each other, and never across. It is the same rule as collecting like terms one level up, and a child who is secure there mostly needs practice keeping three columns straight rather than a new idea.
Why this sheet works
Subtraction is where the marks go. (3x² + 5x − 2) − (x² − 4x + 1) means every term in the second bracket changes sign, including the ones that were already negative, and a child who only negates the first term gets an answer that looks plausible. Rewriting the subtraction as an addition of the opposite is slower and it is worth the time.
How this one works
One question from this sheet, worked through a step at a time.
(6x² + 8x + 3) − (4x² − 7x + 2)
6x² + 8x + 3−4x² − 7x + 2
6x² + 8x + 3+−4x² + 7x − 2
2x² + 15x + 1
2x² + 15x + 1
Only matching terms combine: x² with x², x with x, numbers with numbers. Line the two brackets up by power before doing anything, and nothing can drift into the wrong column.
Subtracting a bracket flips the sign of EVERY term inside it, not just the first: taking away 4x² − 7x + 2 is the same as adding −4x² + 7x − 2. Rewrite it that way before adding — it is slower and it is the step that gets marks.
Now take the columns one at a time, adding the rewritten signs. x²: 6 + (−4) = 2. x: 8 + 7 = 15. Numbers: 3 + (−2) = 1.
So the answer is 2x² + 15x + 1. Read it back and count the terms — three went in on each side, and three come out.
