Distributive Property & Combining Like Terms
Multiplying a sum means multiplying every term inside — no exceptions, no shortcuts. Then combine what matches. These two moves simplify almost every expression you will ever meet.
1 Understand
The core idea in plain language.
The distributive property says a(b + c) = ab + ac: the outside factor multiplies each term inside the parentheses. Think of it as delivering a package to every house on the street — skip one and the delivery is wrong: 3(x + 4) = 3x + 12, not 3x + 4.
Signs ride along with the distribution. In −2(x − 3), the −2 multiplies both x and −3: −2x + 6 (negative × negative = positive). A leading minus is just −1 distributing: −(x + 5) = −x − 5.
Like terms have the same variable and the same exponent: 3x and −7x are like terms; 3x and 3x² are not, and neither are 3x and 4. Only like terms combine: 3x − 7x + 4 = −4x + 4.
A full simplification runs the pipeline: distribute first (clear parentheses), then combine like terms. In 2(x + 3) + 4x − 5: distribute → 2x + 6 + 4x − 5; combine → 6x + 1.
2 See it
Diagrams that make the idea visual.
The area model: 3(x + 4)
A rectangle with height 3 and width (x + 4) splits into two smaller rectangles: 3·x and 3·4.
Total area two ways: 3(x + 4) or 3x + 12 — the distributive property is just “the whole equals the sum of its parts.”
Like terms match exactly
3x and −7x: same variable, same exponent → combine into −4x.
3x² and 4 match nothing — they sit out the round.
3 The key rule
The outside factor multiplies every term inside — including the signs. A leading minus means distribute −1. After distributing, combine like terms: same variable AND same exponent.
4 Worked examples
Follow each step. The pattern is always the same.
- Multiply 3 by each term: 3·x = 3x and 3·4 = 12.
- Result: 3x + 12.
Check: Test x = 2: 3(6) = 18 and 3(2) + 12 = 18. Correct.
- −2 · x = −2x.
- −2 · (−3) = +6 (negative × negative).
- Result: −2x + 6.
Check: Test x = 5: −2(2) = −4 and −2(5) + 6 = −4. Correct.
- Like terms: 3x and −7x (same x¹).
- Combine: 3x − 7x = −4x. The constant 4 has no partner.
- Result: −4x + 4.
Check: Test x = 1: 3 − 7 + 4 = 0 and −4 + 4 = 0. Correct.
- Distribute first: 2(x + 3) = 2x + 6.
- Now 2x + 6 + 4x − 5.
- Combine x-terms: 2x + 4x = 6x; constants: 6 − 5 = 1.
- Result: 6x + 1.
Check: Test x = 10: 2(13) + 40 − 5 = 61 and 6(10) + 1 = 61. Correct.
5 Common mistakes
These errors show up on almost every quiz. Spot them now and they will never cost you points.
6 Key vocabulary
Say these words like you mean them.
- Distributive property
- a(b + c) = ab + ac — multiply the outside factor by every inside term.
- Like terms
- Terms with the same variable and exponent: 3x and −7x.
- Coefficient
- The number multiplying the variable part.
- Constant term
- A term with no variable.
- Simplify
- Distribute to clear parentheses, then combine like terms.
- Expand
- Multiply out using the distributive property.
7 Quick check
Try each one on paper first, then reveal the answer.
Key points to remember
- a(b + c) = ab + ac — every inside term gets multiplied, signs included.
- A leading minus distributes −1: −(x + 5) = −x − 5.
- Like terms need the same variable AND exponent — 3x and 3x² do not combine.
- Pipeline: distribute first, then combine like terms.
- Test with a number (x = 2, x = 10) to verify a simplification instantly.