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

How do you add and subtract polynomials?

To add polynomials, drop the parentheses and combine like terms, meaning terms with the same variable raised to the same power. To subtract, first change the sign of every term in the second polynomial, then combine. For example, (4x² + 3x − 5) − (x² − 2x + 1) becomes 4x² + 3x − 5 − x² + 2x − 1, which simplifies to 3x² + 5x − 6.

What is the rule for adding and subtracting polynomials?

The rule is that you can only combine like terms, and you add or subtract their coefficients while the exponents stay the same. So 5x³ + 2x³ = 7x³, not 7x⁶. Terms such as x², x and plain numbers are different kinds of terms and stay separate. When subtracting, distribute the minus sign across the whole second polynomial before combining anything.

How do you add and subtract polynomials with different exponents?

Terms with different exponents are unlike terms, so they cannot be combined; they simply carry into the answer. Group the terms by power, combine only the matching ones, and write the result from the highest power to the lowest. For example, (2x³ + 4x) + (5x² − x) has no partners for 2x³ or 5x², so the sum is 2x³ + 5x² + 3x.

How is adding and subtracting polynomials like adding and subtracting numbers?

It works much like adding numbers by place value. When you add 342 + 125, you add hundreds to hundreds, tens to tens and ones to ones. With polynomials, the powers of x play the role of place values: x² terms go with x² terms, x terms with x terms, and constants with constants. So (3x² + 4x + 2) + (x² + 2x + 5) = 4x² + 6x + 7, just like 342 + 125 = 467.

What grade is adding and subtracting polynomials?

Most students learn to add and subtract polynomials in Algebra 1, usually in 9th grade or in 8th grade for accelerated students, right after combining like terms and the exponent rules. It comes back in Algebra 2 with higher-degree polynomials. If your student is comfortable simplifying expressions such as 3a + 2b − a, they already have the main skill.

How can adding and subtracting polynomials be used in real life?

Polynomials often model cost, revenue, area and distance, and adding or subtracting them combines those models. For example, if a small business has revenue R = 50x − 0.5x² and cost C = 20x + 300 for x items sold, its profit is R − C = −0.5x² + 30x − 300. Subtracting the polynomials gives one formula that works for any number of items.

Related Topics

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