7.2Adding & Subtracting Polynomials
Combine like terms to add and subtract polynomial expressions. Write results in standard form and apply these skills to geometry, business, and real-world modeling.
Why This Matters
Adding and subtracting polynomials is the foundation for all polynomial algebra. You'll use these skills when combining like terms in Algebra 2, simplifying expressions in Calculus, and working with vector components in Physics.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do we add and subtract polynomials, and why can we only combine terms that are alike?
Lesson Overview
Adding and subtracting polynomials is really about combining like terms. Two terms are like terms if they have the same variable raised to the same exponent — for example, 3x² and −7x² are like terms, but 3x² and 3x are not. When you add polynomials, you simply remove the parentheses and group like terms together. When you subtract, you must first distribute the negative sign (−1) to every term in the second polynomial before combining. The result should always be written in standard form — terms ordered from highest to lowest degree.
Like Terms — Color-Coded Matching
Terms with the same variable AND same exponent are like terms. Only like terms can be combined.
x² terms (Group A)
3x² + 2x² = 5x²
x terms (Group B)
−5x + 4x = −x
constants (Group C)
7 − 3 = 4
Result
5x² − x + 4
Vertical Addition — Column Alignment
Align like terms in columns. Add each column separately.
x² column
3x² + x² = 4x²
x column
5x − 3x = 2x
constant column
−2 + 7 = 5
Subtracting Polynomials — Distributive Property Flowchart
Handling Missing Terms in Vertical Format
When a polynomial is missing a degree, insert a placeholder with coefficient 0 to keep columns aligned.
Without placeholders (confusing):
With placeholders (correct):
Worked Examples
Add: (3x² + 5x − 2) + (x² − 3x + 7)
Remove parentheses: 3x² + 5x − 2 + x² − 3x + 7
Group like terms: (3x² + x²) + (5x − 3x) + (−2 + 7)
Combine coefficients: 4x² + 2x + 5
Subtract: (5x² + 2x − 4) − (2x² − x + 3)
Distribute −1 to the second polynomial: (5x² + 2x − 4) + (−2x² + x − 3)
Remove parentheses: 5x² + 2x − 4 − 2x² + x − 3
Group like terms: (5x² − 2x²) + (2x + x) + (−4 − 3)
Combine: 3x² + 3x − 7
Simplify: 6x³ − 2x² + 4x − 3x³ + 5x² − x + 8
Group like terms: (6x³ − 3x³) + (−2x² + 5x²) + (4x − x) + 8
Combine each group: 3x³ + 3x² + 3x + 8
Add using the horizontal method: (3x² − 4x + 7) + (x² + 6x − 2)
Write side by side: 3x² − 4x + 7 + x² + 6x − 2
Group: (3x² + x²) + (−4x + 6x) + (7 − 2)
Combine: 4x² + 2x + 5
Subtract using the distributive property: (4x² + 3x − 1) − (2x² − 5x + 6)
Distribute −1: (4x² + 3x − 1) + (−2x² + 5x − 6)
Remove parentheses: 4x² + 3x − 1 − 2x² + 5x − 6
Group: (4x² − 2x²) + (3x + 5x) + (−1 − 6)
Combine: 2x² + 8x − 7
Guided Practice
Guided Practice Video: Adding & Subtracting Polynomials
Watch the guided practice walkthrough for adding and subtracting polynomials, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Add: (4x² + 3x − 5) + (2x² − x + 8)
Hint: Remove parentheses, then group and combine like terms: x² terms, x terms, constants.
Subtract: (6x² − 2x + 1) − (3x² + 4x − 7)
Hint: Distribute −1 to every term in the second polynomial first: −(3x² + 4x − 7) = −3x² − 4x + 7.
Simplify: 5x³ + 2x − 3x³ + 4x² − 7x + 1
Hint: Group like terms by degree: x³ terms together, x² terms together, x terms together, constants together.
Add using vertical format: (2x² + 0x − 3) + (x² + 5x + 4)
Hint: Align like terms in columns. The first polynomial has no x term — use 0x as a placeholder.
A rectangle has length (3x + 4) and width (x + 2). Write and simplify an expression for the perimeter.
Hint: P = 2(length) + 2(width). Distribute the 2 first, then combine like terms.
Key Vocabulary
Like Terms
Terms with the same variable raised to the same exponent. Example: 4x² and −9x² are like terms.
Unlike Terms
Terms with different variables or different exponents. Example: 5x and 5x² cannot be combined.
Combining Like Terms
Adding or subtracting the coefficients of like terms while keeping the variable part unchanged.
Distributive Property
a(b + c) = ab + ac. Used when subtracting: −1(2x − 3) = −2x + 3.
Standard Form
A polynomial written with terms in descending order of degree.
Additive Inverse
The opposite of a polynomial; found by multiplying every term by −1.
Simplify
To combine all like terms so no further combining is possible.
Placeholder Term
A term with coefficient 0 used to hold a degree's column in vertical format (e.g., 0x²).
Interactive Practice — 5 Questions
Which pair are like terms?
What is (2x² + 3x − 1) + (x² − 2x + 4)?
What is (5x + 8) − (3x − 2)?
When subtracting (4x² − 3x + 1) − (2x² + x − 5), what does the second polynomial become after distributing −1?
Which is the simplified form of 6x² − 2x + 3x² + 5x − 4?
Independent Practice
Independent Practice
Add and subtract: (a) (x² + 4x − 3) + (2x² − x + 5) (b) (7x² − 3x + 2) − (4x² + x − 5). Show all steps including grouping like terms.
Simplify each expression: (a) 4x² + 3x − 2x² − 5x + 7 (b) 6x³ − x + 2x² − 3x³ + 4x − 9. Write answers in standard form.
Subtract: (a) (2x³ + x² − 4) − (x³ − 3x² + 2) (b) (8x³ − 2x² + x − 5) − (3x³ + x² − 4x + 2). Distribute −1 carefully.
Real-world: (a) A garden has two sections with areas (3x² + 2x) and (x² + 5x − 1). Find the total area. (b) Revenue is (4x² + 100x) and cost is (x² + 60x + 200). Write an expression for profit.
Add three polynomials: (x² + x) + (2x² − 3) + (−x² + 2x + 5). Then find the perimeter of a rectangle with length (2x + 7) and width (x + 3).
Common Mistakes
Combining unlike terms — e.g., adding 3x² and 5x to get 8x³.
Only combine like terms: same variable and same exponent. 3x² + 5x cannot be simplified.
Forgetting to distribute the subtraction sign to every term — e.g., (3x² + 2x) − (x² − 4) = 3x² + 2x − x² − 4.
Distribute the negative to every term: (3x² + 2x) − (x² − 4) = 3x² + 2x − x² + 4 = 2x² + 2x + 4.
Dropping a term when rewriting after distributing the negative.
Write out every term carefully. Use parentheses and distribute the negative sign before combining.
Adding the exponents when combining like terms — e.g., 2x³ + 3x³ = 5x⁶.
When adding like terms, add only the coefficients. The exponent stays the same: 2x³ + 3x³ = 5x³.
Math Tips
Adding: remove parentheses, group like terms, combine coefficients.
Subtracting: distribute −1 to every term of the second polynomial first.
Implied coefficients: x means 1x; −x means −1x. Don't forget these when combining.
Missing degrees: use 0x² as a placeholder in vertical format to keep columns aligned.
Always write your final answer in standard form (highest degree first).