How to Add and Subtract Polynomials? (+FREE Worksheet!)

Adding and subtracting polynomials is an essential Algebra 1 skill that builds directly on combining like terms. Once you understand that polynomials are simply collections of terms, adding and subtracting them becomes a matter of identifying which terms match and working with their coefficients. This guide breaks down the process with clear steps, worked examples, and two video lessons.

What Does It Mean to Add or Subtract Polynomials?

A polynomial is a sum of terms, each consisting of a coefficient multiplied by a variable raised to a non-negative whole-number exponent. Adding polynomials means writing both polynomials together and combining like terms. Subtracting polynomials means distributing the negative sign to every term of the second polynomial before combining like terms.

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How to Add and Subtract Polynomials

Adding Polynomials

Remove the parentheses (addition does not change any signs) and combine like terms.

• \(\color{blue}{(3x^{2} + 2x – 5) + (x^{2} – 4x + 3)}\)
• Drop parentheses: \(\color{blue}{3x^{2} + 2x – 5 + x^{2} – 4x + 3}\)
• Combine: \(\color{blue}{4x^{2} – 2x – 2}\)

Subtracting Polynomials

Distribute the negative sign to every term inside the second set of parentheses — this changes the sign of each term — then combine like terms.

• \(\color{blue}{(5y^{2} – y + 6) – (2y^{2} + 3y – 4)}\)
• \(\color{blue}{\text{ Distribute } -1}\): \(\color{blue}{5y^{2} – y + 6 – 2y^{2} – 3y + 4}\)
• Combine: \(\color{blue}{3y^{2} – 4y + 10}\)

Step-by-Step Summary

1. If adding, drop the parentheses; no sign changes needed.
2. If subtracting, \(\color{blue}{\text{ distribute } -1}\) to every term in the second polynomial; all signs flip.
3. Identify like terms (same variable and same exponent).
4. Add or subtract the coefficients of like terms.
5. Write the result in descending order of degree.

Watch: Adding & Subtracting Polynomials (Concept Lesson)

This Khan Academy lesson walks through the core concept with clear examples:

Adding and Subtracting Polynomials — Worked Examples

Example 1: Add \(\color{blue}{(3x^{2} + 2x – 5) + (x^{2} – 4x + 3)}\).

Drop parentheses and group like terms:
\(\color{blue}{(3x^{2} + x^{2}) + (2x – 4x) + (-5 + 3) = 4x^{2} – 2x – 2}\)

Example 2: Subtract \(\color{blue}{(5y^{2} – y + 6) – (2y^{2} + 3y – 4)}\).

\(\color{blue}{\text{ Distribute } -1}\): \(\color{blue}{5y^{2} – y + 6 – 2y^{2} – 3y + 4}\)
Group and combine: \(\color{blue}{3y^{2} – 4y + 10}\)

Example 3: Add \(\color{blue}{(2a^{3} – 3a^{2} + a) + (a^{3} + a^{2} – 5a + 2)}\).

Group like terms: \(\color{blue}{(2a^{3} + a^{3}) + (-3a^{2} + a^{2}) + (a – 5a) + 2}\)
Combine: \(\color{blue}{3a^{3} – 2a^{2} – 4a + 2}\)

Example 4: Subtract \(\color{blue}{(4x^{2} – 3x + 1) – (x^{2} – 2x + 5)}\).

\(\color{blue}{\text{ Distribute } -1}\): \(\color{blue}{4x^{2} – 3x + 1 – x^{2} + 2x – 5}\)
Combine: \(\color{blue}{3x^{2} – x – 4}\)

More Practice: Step-by-Step Video

This Khan Academy video works through an advanced multi-term example in full detail:

Exercises for Adding and Subtracting Polynomials

Add or subtract as indicated.

1. \(\color{blue}{(2x^{2} + 3x – 1) + (x^{2} – x + 4)}\)
2. \(\color{blue}{(5m^{2} – 2m + 3) – (m^{2} + 4m – 1)}\)
3. \(\color{blue}{(3y^{3} + y – 2) + (y^{2} – 3y + 5)}\)
4. \(\color{blue}{(6x^{2} – x + 2) – (3x^{2} + 2x – 4)}\)
5. \(\color{blue}{(a^{2} + 4a – 3) + (2a^{2} – a + 1)}\)
6. \(\color{blue}{(4t^{3} – t^{2} + 2t) – (t^{3} + 3t^{2} – t)}\)

Answers

1. \(\color{blue}{3x^{2} + 2x + 3}\)
2. \(\color{blue}{4m^{2} – 6m + 4}\)
3. \(\color{blue}{3y^{3} + y^{2} – 2y + 3}\)
4. \(\color{blue}{3x^{2} – 3x + 6}\)
5. \(\color{blue}{3a^{2} + 3a – 2}\)
6. \(\color{blue}{3t^{3} – 4t^{2} + 3t}\)

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Free Adding and Subtracting Polynomials Worksheet

Ready to practice on your own? Download our free Adding and Subtracting Polynomials worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again — steady practice is the surest way to master Adding and Subtracting Polynomials before a quiz or test.

Download Adding and Subtracting Polynomials Worksheet

Frequently Asked Questions

Why do signs change when subtracting polynomials?

Subtracting a polynomial is the same as adding its opposite. Distributing the negative sign reverses every term’s sign in the subtracted polynomial: \(\color{blue}{-(2y + 3) = -2y – 3}\).

Can you add polynomials with different numbers of terms?

Yes. Simply bring all terms together and combine any that are like terms. Terms with no matching partner stay as they are in the final answer.

What is the degree of the result when you add polynomials?

The degree of the result is at most the highest degree of either polynomial. If the leading terms cancel each other out, the degree of the result may be lower.

Related Topics

• Simplifying Polynomials
• Multiplying Monomials
• Multiplying a Polynomial and a Monomial
• Multiplying Binomials
• Factoring Trinomials