Adding and Subtracting Polynomials- Complete Notes and Examples

What Polynomials Actually Are

A polynomial is just a sum of terms with variables raised to whole number powers. That's it. No tricks. The terms get combined when you add or subtract them, but only like terms can merge together.

Like terms share the exact same variable part. 3x² and -5x² are like terms. 3x² and 3x are not. The exponent matters.

Adding Polynomials: Two Methods

You can add polynomials by stacking them vertically or combining horizontally. Both work. Pick whichever doesn't make you stare blankly at your paper.

Horizontal Method

Group the like terms and add their coefficients. Keep the variable part unchanged.

Example: Add (2x² + 3x + 1) + (4x² - 2x + 5)

Combine the x² terms: 2x² + 4x² = 6x²
Combine the x terms: 3x + (-2x) = x
Combine the constants: 1 + 5 = 6

Answer: 6x² + x + 6

Vertical Method

Line up the terms by degree, then add down each column. Works best when you have multiple variables or want to catch mistakes.

Example:

    2x² + 3x + 1
  + 4x² - 2x + 5
  ---------------
    6x² + x + 6

Same answer. Shocking.

Subtracting Polynomials: The Distributive Property Bites Back

Here's where students mess up. When you subtract a polynomial, you must distribute the negative sign to every term. Not just the first one. Every. Single. One.

Example: Subtract (5x² + 3x - 2) - (2x² - 4x + 7)

Step 1: Distribute the negative
5x² + 3x - 2 - 2x² + 4x - 7

Step 2: Combine like terms
5x² - 2x² = 3x²
3x + 4x = 7x
-2 - 7 = -9

Answer: 3x² + 7x - 9

The subtraction turns the -4x into +4x and the +7 into -7. Forgetting this is the #1 reason polynomial subtraction goes wrong.

Vertical Subtraction

Stack them, then subtract each term going down. If a coefficient is missing, treat it as 0.

Example:

    5x² + 3x - 2
  - 2x² - 4x + 7    ← signs flipped
  ---------------
    3x² + 7x - 9

Notice the signs in the bottom row flipped. That's not optional.

Adding and Subtracting Polynomials with Multiple Variables

Same rules apply. Just track all variables and their exponents.

Example: Add (3xy + 2x² - y) + (5xy - 3x² + 4y)

3xy + 5xy = 8xy
2x² - 3x² = -x²
-y + 4y = 3y

Answer: 8xy - x² + 3y

You cannot combine xy and x² terms. Different variables or different exponents mean they stay separate.

Common Mistakes to Avoid

Forgetting to distribute the negative sign when subtracting. It ruins everything downstream.
Combining unlike terms. x² and x look similar but they are not like terms. Stop trying to merge them.
Dropping signs. Write every sign explicitly. Don't assume a term is positive just because it looks bare.
Misaligning in vertical format. Make sure same-degree terms line up vertically or you'll get garbage answers.

How To: Adding and Subtracting Polynomials in 4 Steps

Here's the process that actually works:

Identify like terms. Scan both polynomials for matching variable parts. Circle or highlight them if needed.
For subtraction, distribute the negative. Flip the sign of every term in the polynomial being subtracted.
Group like terms together. Rearrange mentally or on paper so identical terms are adjacent.
Add or subtract the coefficients. Keep the variable part exactly as it was. Write the result.

Practice Problem: Simplify (4x³ + 2x² - x + 3) - (x³ - 5x² + 4x - 1)

Step 1: Distribute the negative
4x³ + 2x² - x + 3 - x³ + 5x² - 4x + 1

Step 2: Combine
4x³ - x³ = 3x³
2x² + 5x² = 7x²
-x - 4x = -5x
3 + 1 = 4

Answer: 3x³ + 7x² - 5x + 4

Quick Reference: Methods Compared

Method	Best For	Drawback
Horizontal (combine like terms)	Simple expressions, 2-3 terms	Can get messy with many terms
Vertical (column alignment)	Multiple variables, long polynomials	Requires careful alignment
Distribute-then-combine	Subtraction problems	Easy to miss a sign

Most instructors accept either method on tests. If one keeps giving you wrong answers, use the other one instead. There's no style points in polynomial arithmetic.

When You're Stuck

If you're getting a wrong answer, check these in order:

Did you distribute the negative sign to every term when subtracting?
Are you only combining terms with identical variable parts?
Did any term's sign change accidentally during rearrangement?
Did you copy the problem correctly? Half of "wrong answers" are just transcription errors.

Go back to the original problem and verify each term made it to your work unchanged. Then check your arithmetic on the coefficients.