Lesson 2.5

Adding & Subtracting Polynomials

Now that we know what polynomials are and how they behave, it's time to start operating on them. Addition is intuitive — subtraction is where the traps are.

Introduction

Adding and subtracting polynomials is really just combining like terms — a skill you already have from Algebra 1. The only new challenge is distributing the negative sign when subtracting.

Past Knowledge

You already combine like terms: .

Today's Goal

Add and subtract entire polynomials, distributing negatives correctly.

Future Success

This is essential before multiplying and dividing polynomials in the next lessons.

Key Concepts

1. Like Terms

Terms with the same variable raised to the same power are "like terms" and can be combined.

and

are like terms

and

are NOT like terms

2. Adding Polynomials

Remove parentheses and combine like terms. That's it!

↓

3. Subtracting Polynomials

Distribute the negative to every term in the second polynomial, then combine.

↓ Distribute the negative

↓ Combine like terms

💡 The Big Rule

When you subtract a polynomial, every sign in the second set of parentheses flips. Positive becomes negative, and negative becomes positive.

Worked Examples

Example 1: Adding Polynomials

Basic

Simplify .

Group Like Terms

Combine

Example 2: Subtracting Polynomials

Intermediate

Simplify .

Distribute the Negative

Change the sign of every term in the second polynomial.

Combine Like Terms

terms:

Constants:

Example 3: Multi-Step Operation

Advanced

If and , find .

Set Up the Subtraction

Distribute — Watch the Double Negative!

becomes .

Combine and Write in Standard Form

Common Pitfalls

Only Subtracting the First Term

Students often write as . The negative must be distributed to ALL terms: .

Combining Unlike Terms

≠ . You can only combine terms with the exact same variable and exponent. These are unlike terms and stay separate.

Real-Life Applications

In business, profit is Revenue minus Cost: . If both revenue and cost are modeled by polynomials, finding the profit function is literally subtracting polynomials. Getting the signs wrong means the entire financial model is wrong!