Polynomial Addition and Subtraction Formula

The Formula

(P + Q)(x) = P(x) + Q(x), combining like terms: ax^n + bx^n = (a+b)x^n

When to use: Think of like terms as the same type of object: 3x^2 and 5x^2 are both 'x^2 things,' so you can combine them into 8x^2, just like 3 apples plus 5 apples equals 8 apples. You cannot combine x^2 and x any more than you can add apples and oranges.

Quick Example

(3x^2 + 2x - 5) + (x^2 - 4x + 7) = 4x^2 - 2x + 2 — combine x^2, x, and constant terms separately.

Notation

Like terms share the same variable and exponent. Align terms by degree when adding vertically. The minus sign in subtraction distributes to every term.

What This Formula Means

Adding or subtracting polynomials by combining like terms—terms with the same variable raised to the same power.

Think of like terms as the same type of object: 3x^2 and 5x^2 are both 'x^2 things,' so you can combine them into 8x^2, just like 3 apples plus 5 apples equals 8 apples. You cannot combine x^2 and x any more than you can add apples and oranges.

Formal View

For P(x) = \sum_{k} a_k x^k and Q(x) = \sum_{k} b_k x^k in \mathbb{R}[x]: (P \pm Q)(x) = \sum_{k} (a_k \pm b_k) x^k. The degree satisfies \deg(P + Q) \leq \max(\deg P, \deg Q).

Worked Examples

Example 1

easy
Add (3x^2 + 5x - 2) + (x^2 - 3x + 7).

Solution

  1. 1
    Step 1: Group like terms: (3x^2 + x^2) + (5x - 3x) + (-2 + 7).
  2. 2
    Step 2: Combine: 4x^2 + 2x + 5.
  3. 3
    Check: At x = 1: (3+5-2) + (1-3+7) = 6 + 5 = 11 and 4+2+5 = 11 ✓

Answer

4x^2 + 2x + 5
To add polynomials, combine like terms — terms with the same variable raised to the same power. The coefficients are added while the variable parts stay the same.

Example 2

medium
Subtract (5x^3 - 2x^2 + x) - (3x^3 + x^2 - 4x + 2).

Common Mistakes

  • Forgetting to distribute the negative sign when subtracting: (3x - 2) - (x + 5) \neq 3x - 2 - x + 5
  • Combining unlike terms such as adding x^2 and x together
  • Dropping terms when rewriting—always account for every term in both polynomials

Why This Formula Matters

Combining like terms is the most fundamental simplification skill in algebra and appears in every area of higher math.

Frequently Asked Questions

What is the Polynomial Addition and Subtraction formula?

Adding or subtracting polynomials by combining like terms—terms with the same variable raised to the same power.

How do you use the Polynomial Addition and Subtraction formula?

Think of like terms as the same type of object: 3x^2 and 5x^2 are both 'x^2 things,' so you can combine them into 8x^2, just like 3 apples plus 5 apples equals 8 apples. You cannot combine x^2 and x any more than you can add apples and oranges.

What do the symbols mean in the Polynomial Addition and Subtraction formula?

Like terms share the same variable and exponent. Align terms by degree when adding vertically. The minus sign in subtraction distributes to every term.

Why is the Polynomial Addition and Subtraction formula important in Math?

Combining like terms is the most fundamental simplification skill in algebra and appears in every area of higher math.

What do students get wrong about Polynomial Addition and Subtraction?

When subtracting, the minus sign must be distributed to EVERY term in the second polynomial, not just the first.

What should I learn before the Polynomial Addition and Subtraction formula?

Before studying the Polynomial Addition and Subtraction formula, you should understand: expressions, polynomials.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Polynomial Long Division: Step-by-Step Method with Examples →
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