How to Add and Subtract Polynomials: A Step-by-Step Guide

Introduction to Polynomial Addition and Subtraction

Adding and subtracting polynomials is a foundational skill in algebra that combines like terms to simplify expressions. This guide explains how to approach these operations methodically so you can apply them in equations, graphing, and more advanced math. You will learn to identify like terms, align terms by degree, and handle subtraction by distributing negative signs. The steps below align with practices often found in Khan Academy exercises and instructional videos, helping you build accuracy and confidence.

What Are Polynomials

A polynomial is an expression made of variables and constants combined using addition, subtraction, and multiplication, with non-negative integer exponents. Examples include monomials (one term), binomials (two terms), and trinomials (three terms). When adding or subtracting polynomials, the key is to work with like terms—terms that have the exact same variable parts. The degree of a polynomial is the highest exponent of its variable in standard form.

Step-by-Step: Adding Polynomials

1) Remove Parentheses

Start by removing any parentheses. If there is a plus sign in front, the terms inside remain unchanged. In most practice problems, parentheses are used mainly for grouping.

2) Identify Like Terms

Like terms share the same variables raised to the same powers. For example, 3x^2 and -5x^2 are like terms, as are 4y and 7y. Constants such as 6 and -2 are also like terms.

3) Group and Combine

Rearrange terms using the commutative property of addition so that like terms are next to each other. Then add their coefficients while keeping the variable part unchanged. For instance, 3x^2 + 2x^2 becomes 5x^2.

4) Write in Standard Form

Write the resulting polynomial with terms in descending order of degree. This standard form makes it easier to compare and work with multiple expressions.

Step-by-Step: Subtracting Polynomynomials

1) Distribute the Negative Sign

Subtraction is equivalent to adding the opposite. Replace subtracting a polynomial with adding its opposite, which means changing the sign of every term inside the parentheses.

2) Remove Parentheses

Once the subtraction has been rewritten as addition, drop the parentheses and proceed carefully to keep track of signs.

3) Group Like Terms

Rearrange the expression so that like terms are aligned. This step helps avoid errors when combining coefficients.

4) Combine and Simplify

Add the coefficients of like terms, and write the simplified polynomial in standard form. Double-check that each term’s sign is correct after distribution.

Common Mistakes and How to Avoid Them

• Forgetting to change signs when subtracting a polynomial. Always distribute the negative to every term inside the parentheses.
• Combining terms that are not like terms, such as x^2 and x. Only terms with identical variable parts can be combined.
• Skipping the standard form step, which can make it harder to compare expressions or identify degrees later.

Examples with Solutions

Example 1 — Addition: (2x^2 + 3x - 4) + (x^2 - 5x + 6)

Remove parentheses: 2x^2 + 3x - 4 + x^2 - 5x + 6

Group like terms: (2x^2 + x^2) + (3x - 5x) + (-4 + 6)

Combine: 3x^2 - 2x + 2

Example 2 — Subtraction: (4y^2 - y + 5) - (2y^2 + 3y - 1)

Distribute the negative: 4y^2 - y + 5 - 2y^2 - 3y + 1

Group like terms: (4y^2 - 2y^2) + (-y - 3y) + (5 + 1)

Combine: 2y^2 - 4y + 6

Practice Tips and Next Steps

To build fluency, practice rewriting each subtraction problem as addition of the opposite before simplifying. Work slowly when distributing negatives, and check that each term’s sign is correct. Use vertical alignment to keep like terms lined up, especially with longer expressions. As you advance, these skills will support operations with factoring, solving equations, and working with rational expressions.

Quick Reference: Adding vs Subtracting Polynomials