HighSchoolQ

➕ Algebra 1 · Equations

The Distributive Property: notes and common mistakes

Study notes

Key ideas

  • a(b + c) = ab + ac: the outside number multiplies every term inside the parentheses.
  • Each term carries its sign. x − 7 is x + (−7), so a negative outside number changes the sign of every term inside.
  • The property also works backward. Pulling a common factor out front is called factoring: ab + ac = a(b + c).
  • It applies to a number or a term times a sum or difference. An exponent on a sum does not distribute, and neither does dividing a number by a sum.

Steps

  1. Draw an arrow from the outside number to each term inside (counting each term’s sign).
  2. Multiply the outside number by each term, one at a time.
  3. Write the products, then combine any like terms.
  4. Check by trying a test number such as x = 3 (not 0, 1 or 2) in the original and in your answer.

Formulas

  • Distributing over addition: a(b + c) = ab + ac
  • Distributing over subtraction: a(b − c) = ab − ac
  • Parentheses first: (b + c)a = ba + ca
  • Factoring (backward): ab + ac = a(b + c)
  • Minus sign in front: −(b + c) = −b − c
  • Dividing a sum by a number: (b + c) ÷ a = (b ÷ a) + (c ÷ a) (when a ≠ 0)

Watch out

  • Multiply every term. 4(x + 6) is not 4x + 6.
  • A negative outside number changes the sign of every term inside: −2(x − 7) = −2x + 14, not −2x − 14.
  • A term outside the parentheses is not part of the distribution. Multiply first, then add: 9 + 2(x + 4) = 2x + 17.
  • An exponent does not distribute over a sum: (x + 3)2 ≠ x2 + 9.

Why it works

  • Why does it work? 4(x + 6) means 4 groups of (x + 6). Each group has one x and one 6, so you end up with 4 x’s and 4 sixes.
  • Why use it? It lets you take parentheses apart so you can combine like terms and, soon, solve equations that have parentheses in them.

Common mistakes

✗ Mistake9(y + 2) = 9y + 2 (only the first term gets multiplied).
✓ InsteadThe 9 multiplies every term: 9(y + 2) = 9y + 18.
✗ Mistake−8(x − 5) = −8x − 40 (the second sign is left alone).
✓ InsteadThe second term is −5, and (−8) × (−5) = +40. So −8(x − 5) = −8x + 40.
✗ Mistake−(3x − 10) = −3x − 10 (only the first sign flips).
✓ InsteadThe minus sign multiplies every term by −1, so both signs flip: −(3x − 10) = −3x + 10.
✗ Mistake(x + 3)2 = x2 + 9 (distributing an exponent).
✓ InsteadAn exponent does not distribute over a sum. (x + 3)2 means (x + 3)(x + 3), which equals x2 + 6x + 9. You will learn how in Multiplying Polynomials.
✗ Mistake6 + 5(x + 1) = 11(x + 1) (adding before multiplying).
✓ InsteadMultiplication comes before addition. Distribute first: 6 + 5x + 5 = 5x + 11.

Teach it back

Explain to a friend (or your dog!) how to expand 3(4x − 7), why the −7 matters, and how you can check your answer.

Explaining it out loud shows what you really know. Did you…

  • I said the 3 multiplies every term, including the −7.
  • I got 12x − 21 and explained why the sign stays negative.
  • I checked with a test number such as x = 3: 3(4 × 3 − 7) = 15 and 12 × 3 − 21 = 15.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

Downloads a calendar file (.ics). Works with Google, Apple and Outlook calendars. Nothing is sent to us.