How to Add and Subtract Polynomials: Step by Step Guide

Many Algebra 1 students struggle with polynomials—confusing like and unlike terms, forgetting that constants are like terms, making sign errors, or not knowing what a coefficient, constant, or variable really means. Our step-by-step guide and downloadable worksheets give students clear practice to build confidence and avoid these common mistakes.

What Are Polynomials?

A polynomial is an algebraic expression made by adding, subtracting, or multiplying numbers and variables, with exponents that are whole numbers. It has a limited number of terms. Each term can have constants (numbers), variables, or both, multiplied together.

Example:

5𝑥² + 3𝑥 − 7

This polynomial has three terms:

• The first is 5𝑥², in this term:
  • The coefficient is 5 (the number in front);
  • The variable is 𝑥;
  • The exponent 2 tells us it’s a squared term.

• The second is 3𝑥, in this term:
  • The coefficient is 3;
  • The variable is 𝑥;
  • The exponent is 1 (𝑥 = 𝑥¹).

• The last is -7, in this term:
  • It has no variable;
  • It is a constant.

What Are Like Terms?

Like terms are terms that have the same variables raised to the same powers. We can only combine like terms when adding or subtracting polynomials.

For example, in:

(5𝑥² + 3𝑥 − 7) and (2𝑥² − 4𝑥 + 6),

• 5𝑥² and 2𝑥² are like terms (both have 𝑥²)
• 3𝑥 and −4𝑥 are like terms (both have 𝑥)
• −7 and 6 are like terms (both constants)

How to Add Polynomials?

Example:

(5𝑥² + 3𝑥 − 7) + (2𝑥² − 4𝑥 + 6)

To add polynomials, remove the parentheses first.

5𝑥² + 3𝑥 − 7 + 2𝑥² − 4𝑥 + 6

Then combine like terms:

• 5𝑥² + 2𝑥² = 7𝑥²
• 3𝑥 − 4𝑥 = −𝑥
• −7 + 6 = −1

Final result:

(5𝑥² + 3𝑥 − 7) + (2𝑥² − 4𝑥 + 6) = 7𝑥² − 𝑥 − 1

How to Subtract Polynomials

Example:

(5𝑥² + 3𝑥 − 7) − (2𝑥² − 4𝑥 + 6)

To do the subtraction, remove the parentheses first.

5𝑥² + 3𝑥 − 7 − 2𝑥² + 4𝑥 − 6

Then, combine like terms:

• 5𝑥² − 2𝑥² = 3𝑥²
• 3𝑥 + 4𝑥 = 7𝑥
• −7 − 6 = −13

Final result:

(5𝑥² + 3𝑥 − 7) − (2𝑥² − 4𝑥 + 6) = 3𝑥² + 7𝑥 − 13

Example Problems: Addition and Subtraction of Polynomials

Example 1

Problem: What is the result of (5𝑟𝑡 − 3𝑟𝑤 − 8𝑡𝑤) + (6𝑟𝑡 − 4𝑟𝑤 + 2𝑡𝑤)?

Answer:

Group like terms:

• 5𝑟𝑡 + 6𝑟𝑡 = 11𝑟𝑡
• −3𝑟𝑤 − 4𝑟𝑤 = −7𝑟𝑤
• −8𝑡𝑤 + 2𝑡𝑤 = −6𝑡𝑤

Final result: (5𝑟𝑡 − 3𝑟𝑤 − 8𝑡𝑤) + (6𝑟𝑡 − 4𝑟𝑤 + 2𝑡𝑤) = 11𝑟𝑡 − 7𝑟𝑤 − 6𝑡𝑤

Example 2

Problem: What is (4𝑎² + 5𝑎 − 3) − (2𝑎² − 6𝑎 + 4)?

Answer:

First, distribute the negative:

4𝑎² + 5𝑎 − 3 − 2𝑎² + 6𝑎 − 4

Then, group like terms:

• 4𝑎² − 2𝑎² = 2𝑎²
• 5𝑎 + 6𝑎 = 11𝑎
• −3 − 4 = −7

Final result: (4𝑎² + 5𝑎 − 3) − (2𝑎² − 6𝑎 + 4) = 2𝑎² + 11𝑎 − 7

Summary: Addition and Subtraction of Polynomials

• Polynomials are made up of terms with variables, constants, and exponents.
• Like terms have the same variables and exponents. Constants are also like terms, because they have no variables.
• To add or subtract, group and combine like terms.
• Always pay attention to signs, especially when subtracting.

Additional Math Topics for Algebra 1 Polynomials – with Free Worksheets

• How to Multiply Polynomials: From 1 Term to 3 Terms
• Polynomial Multiplication in Algebra: Formulas, Examples, and AMC 10 Applications
• How to Divide a Polynomial by a Monomial in 2 Simple Steps
• How to Divide Polynomials in 4 Simple Steps

Want more printable practice?

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