Add and Subtract Polynomials- A Step-by-Step Guide

What Are Polynomials?

A polynomial is a mathematical expression with multiple terms added or subtracted together. Each term contains a coefficient (the number in front) and a variable raised to a power. The variable itself never goes underground—it's always on top, positive.

Examples:

3x² + 2x - 5
4y³ - 3y² + 7y + 1
2a + 6

The degree of a polynomial is the highest exponent. A polynomial with degree 2 is quadratic. Degree 3 is cubic. Anything higher just gets called by its degree number.

Adding Polynomials

Adding polynomials means combining like terms. Like terms are terms with the exact same variable part—the same letter and the same exponent.

You can only add 3x² and 5x² together. You cannot add 3x² and 5x³. They're incompatible. The variables don't match.

How to Add Polynomials

Step 1: Write both polynomials clearly, preferably with parentheses.

Step 2: Remove parentheses. If there's a negative sign in front, distribute it.

Step 3: Identify like terms by matching variables and exponents.

Step 4: Add the coefficients of like terms.

Step 5: Write the result, keeping variables and exponents exactly as they were.

Example: Adding Polynomials

Add (3x² + 2x - 4) + (5x² - 3x + 7)

First, remove the parentheses:

3x² + 2x - 4 + 5x² - 3x + 7

Now group like terms:

3x² + 5x² = 8x²

2x - 3x = -x

-4 + 7 = 3

Final answer: 8x² - x + 3

Subtracting Polynomials

Subtraction is where people mess up. The problem is the negative sign in front of the second polynomial. You have to distribute that negative to every term inside the parentheses before you combine anything.

This is the step most people skip. Then they wonder why their answer is wrong.

How to Subtract Polynomials

Step 1: Write the problem with both polynomials.

Step 2: Distribute the negative sign to every term in the second polynomial. This changes all the plus signs to minus and all the minus signs to plus.

Step 3: Combine like terms.

Step 4: Simplify.

Example: Subtracting Polynomials

Subtract (4x³ - 2x² + 6) - (3x³ + 5x² - 4)

Distribute the negative sign:

4x³ - 2x² + 6 - 3x³ - 5x² + 4

Notice what happened: the plus became minus, and the minus became plus.

Now combine like terms:

4x³ - 3x³ = x³

-2x² - 5x² = -7x²

6 + 4 = 10

Final answer: x³ - 7x² + 10

Vertical vs. Horizontal Method

You can line polynomials up vertically and add or subtract column by column. Some people find this easier because it forces you to align like terms correctly.

Vertical Addition Example

    3x² + 4x + 5
+   2x² -  x - 3
----------------
    5x² + 3x + 2

Vertical Subtraction Example

    5x² + 6x - 2
-   3x² + 2x + 4
----------------
    2x² + 4x - 6

The vertical method works well when polynomials have the same terms. When they don't, you need to leave blanks for missing terms or use the horizontal method.

Adding vs. Subtracting Polynomials

Operation	Key Rule	Common Mistake
Addition	Combine like terms only	Trying to combine unlike terms (x² + x)
Subtraction	Distribute negative first	Forgetting to distribute the negative sign

Common Mistakes to Avoid

Distributing the negative incorrectly: If you see a minus sign before parentheses, it flips every sign inside. Every single one.
Combining unlike terms: x² and x are not the same thing. Keep them separate.
Dropping terms: Write out every term when removing parentheses. Don't lose anything in the process.
Forgetting the coefficient: A term like "x²" has an implied coefficient of 1. A term like "-x²" has a coefficient of -1.

Practice: Add and Subtract These

Try these problems. Answers below.

1. (2x + 3) + (5x - 7) = ?

2. (6y² - 4y + 1) + (2y² + 8y - 3) = ?

3. (4a³ - 2a² + a) - (3a³ + a² - 5a) = ?

4. (x² + 3x - 9) - (2x² - x + 4) = ?

Answers

1. 7x - 4

2. 8y² + 4y - 2

3. a³ - 3a² + 6a

4. -x² + 4x - 13

Getting Started: The Checklist

Before you submit any polynomial problem:

✅ Did you remove all parentheses?
✅ Did you distribute every negative sign correctly?
✅ Did you group all like terms together?
✅ Did you add or subtract only the coefficients, not the variables?
✅ Is your answer written in standard form (highest degree first)?

If you answered yes to all five, your answer is probably correct. If you skipped step 2, start over. That's where most errors happen.