Pengi Editor's Note: This article was originally published by Think Academy. We're sharing it here for educational value. Think Academy is a leading K-12 math education provider.
Addition and Subtraction of Polynomials: Step-by-Step Guide
Adding and subtracting polynomials is a foundational algebra skill. The key is identifying like terms — terms with the same variable and the same exponent — and combining only those.
What Are Polynomials?
A polynomial is an algebraic expression with one or more terms, where each term consists of a coefficient and a variable raised to a non-negative integer exponent.
Examples:
- 3x² + 2x − 5 (three terms = trinomial)
- 4x³ − 7x (two terms = binomial)
- 9 (one term = monomial)
What Are Like Terms?
Like terms have the same variable raised to the same power.
Like terms: 3x² and −7x² (both are x² terms)
Not like terms: 3x² and 5x (different exponents), 3x and 3y (different variables)
You can only add or subtract like terms.
Adding Polynomials
To add polynomials: remove parentheses and combine like terms.
Example 1
Add: (3x² + 2x − 1) + (x² − 5x + 4)
Step 1: Remove parentheses
= 3x² + 2x − 1 + x² − 5x + 4
Step 2: Group like terms
= (3x² + x²) + (2x − 5x) + (−1 + 4)
Step 3: Combine
= 4x² − 3x + 3
Example 2
Add: (2x³ − x + 6) + (−3x³ + 4x² + 2x − 1)
= 2x³ − x + 6 − 3x³ + 4x² + 2x − 1
= (2x³ − 3x³) + 4x² + (−x + 2x) + (6 − 1)
= −x³ + 4x² + x + 5
Subtracting Polynomials
To subtract polynomials: distribute the negative sign (change all signs in the second polynomial) and combine like terms.
Example 3
Subtract: (5x² + 3x − 2) − (2x² − x + 4)
Step 1: Distribute the negative sign
= 5x² + 3x − 2 − 2x² + x − 4
Step 2: Combine like terms
= (5x² − 2x²) + (3x + x) + (−2 − 4)
= 3x² + 4x − 6
Example 4 — Common Mistake Demonstration
Subtract: (4x + 7) − (3x − 5)
Incorrect approach (forgetting to distribute the negative):
= 4x + 7 − 3x − 5 ← WRONG
Correct approach:
= 4x + 7 − 3x + 5 ← The −5 becomes +5
= x + 12 ✓
Adding and Subtracting with Multiple Variables
Example 5
Simplify: (3x²y + 2xy² − 5) + (x²y − 4xy² + 3)
= (3x²y + x²y) + (2xy² − 4xy²) + (−5 + 3)
= 4x²y − 2xy² − 2
The Column Method (Vertical Addition)
For longer polynomials, arranging terms in columns helps organize the work.
Example: Add (3x³ + x² − 2x + 1) and (−x³ + 4x² + 3x − 5)
3x³ + x² − 2x + 1
+ (−x³) + 4x² + 3x − 5
─────────────────────────
2x³ + 5x² + x − 4
Key Rules Summary
| Operation | Rule |
|---|---|
| Adding | Remove parentheses; combine like terms |
| Subtracting | Distribute the negative; combine like terms |
| Like terms | Same variable AND same exponent |
Practice Problems
- (4x² + 3x − 2) + (−2x² + x + 5)
- (6x² − x + 3) − (2x² + 4x − 1)
- (x³ + 2x² − x) + (−x³ + x − 7)
- (5x + 3y − 2) − (2x − y + 4)
Want more printable practice?
Explore the K–12 Worksheets Hub
Try Pengi AI — Smarter Math Practice for Students
Pengi AI supports K–12 learners with personalized math practice, guided explanations, and feedback designed to help them build confidence and improve steadily.
