➕ Algebra 1 · Equations
The Distributive Property
Learn the distributive property, a(b + c) = ab + ac, with area models, negatives, factoring and mental math, plus memory tricks, games and practice.
About standards and checking
Written by AI, then independently re-solved and checked by separate AI reviewers and automated answer checks. Written by Claude (Sonnet 5.5) from the concept template. Automated gate: schema, notation and every math answer verified by evaluated expressions. Three blind re-solves by separate AI solvers with no access to the answer key: every keyed answer matched. Two prose audits by a separate, stronger AI reviewer (Claude Opus 5.5): every finding fixed, no wrong answer keys found. Final small edits re-checked by the automated gate (no human teacher review yet). Last checked 2026-11-07.
- 6.EE.A.3: Apply the properties of operations to generate equivalent expressions.
- 7.EE.A.1: Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
By the end you can…
- Use a(b + c) = ab + ac to expand expressions like 4(x + 6), including ones with subtraction and negative numbers.
- Explain why the property works using groups and a rectangle (area model).
- Use the property backward to factor out a common factor, and to do mental math.
- Spot where the property does not apply, such as a power of a sum like (x + 3)2.
Lesson (school style)
To distribute means to hand something out to everyone. The distributive property says that when a number or term multiplies a sum or difference in parentheses, it multiplies every term inside: a(b + c) = ab + ac. It lets you take parentheses apart. Read backward, it lets you pull a common factor out front.
What the property says
a(b + c) = ab + ac and a(b − c) = ab − ac.
The number outside the parentheses multiplies each term inside, one at a time. Then you combine the products. (When the outside number is negative, take extra care with the signs. That comes up in a moment.)
Try it with plain numbers: 3(4 + 5) = 3 × 9 = 27, and 3 × 4 + 3 × 5 = 12 + 15 = 27. The same answer both ways.
It also works when the parentheses come first: (b + c)a = ba + ca.
Why it works: groups and rectangles
4(x + 6) means 4 groups of (x + 6). Each group has one x and one 6, so altogether you have 4 x’s and 4 sixes: 4x + 24.
A rectangle shows the same thing. Picture a rectangle 4 tall and x + 6 wide. Cut it into a part that is x wide and a part that is 6 wide. The parts have areas 4x and 4 × 6 = 24, and together they fill the whole rectangle.
The picture needs x to be a positive length, but the rule itself is true for every number, including negatives and fractions.
Subtraction and negative numbers
Every term carries its own sign. In x − 7 the second term is −7 (subtracting 7 is the same as adding −7). So the outside number multiplies the sign too.
- Positive outside: 5(2x − 3) = 10x − 15. The minus stays a minus.
- Negative outside: −2(x − 7) = −2x + 14. Multiply the signs: (−2) × (−7) = +14.
- A minus sign with no number in front of the parentheses means −1: −(4x + 9) = −4x − 9. Every sign inside flips.
Reminder: same signs multiply to a positive, and different signs multiply to a negative.
Where it applies, and where it does not
The property works when the parentheses hold a sum or difference: terms joined by + or −.
- A term sitting outside the parentheses just waits, because multiplication comes before addition: 9 + 2(x + 4) = 9 + 2x + 8 = 2x + 17.
- An exponent does not distribute over a sum. (x + 3)2 means (x + 3)(x + 3), and it is not x2 + 9. Try x = 4: (4 + 3)2 = 49, but 42 + 9 = 25.
- Dividing a sum by a number (not 0) does work: (6x + 9) ÷ 3 = 2x + 3. Dividing a number by a sum does not: 24 ÷ (2 + 4) = 4, but 24 ÷ 2 + 24 ÷ 4 = 12 + 6 = 18.
Run it backward: factoring
Read the property from right to left and you get factoring: ab + ac = a(b + c). To factor 8x + 20:
- Find a number that divides every term. Both 8 and 20 are divisible by 4, and nothing larger works for both.
- Write it outside the parentheses and put what is left of each term inside: 4(2x + 5).
- Check by distributing: 4 × 2x = 8x and 4 × 5 = 20.
Using the largest common factor finishes the job. Using 2 would give 2(4x + 10), which is still true but not fully factored.
A mental-math trick
Split a hard number into easy pieces. 6 × 47 = 6(50 − 3) = 300 − 18 = 282. You could also split it as 6(40 + 7) = 240 + 42 = 282.
It is the same property, only with numbers instead of variables.
Key words
- Distributive property
- The rule that a number or term multiplying a sum or difference multiplies each term inside the parentheses: a(b + c) = ab + ac.
- Term
- A part of an expression that is separated from the other parts by + or −. The sign in front belongs to the term: in 4x − 24 the terms are 4x and −24.
- Expand
- To rewrite a product like 4(x + 6) as a sum or difference like 4x + 24 by distributing.
- Factor (verb)
- To rewrite a sum like 8x + 20 as a product like 4(2x + 5) by pulling out a common factor.
- Equivalent expressions
- Expressions that have the same value for every value of the variable, like 4(x + 6) and 4x + 24.
- Greatest common factor
- The largest number that divides every term in a sum. For 8x + 20 it is 4.
Worked examples
Example 1. Expand 4(x + 6).
- The 4 must multiply every term inside the parentheses: the x and the 6.
- 4 × x = 4x and 4 × 6 = 24.
- Join the results with the plus sign that was inside: 4x + 24.
- Check with x = 3: 4(3 + 6) = 4 × 9 = 36, and 4(3) + 24 = 12 + 24 = 36. Both match.
Answer: Answer: 4x + 24
Example 2. Expand 5(2x − 3).
- Think of 2x − 3 as 2x + (−3). The 5 multiplies both terms.
- 5 × 2x = 10x and 5 × (−3) = −15.
- Put them together: 10x − 15.
- Check with x = 4: 5(2 × 4 − 3) = 5 × 5 = 25, and 10 × 4 − 15 = 40 − 15 = 25. Both match.
Answer: Answer: 10x − 15
Example 3. Expand −2(x − 7).
- The outside number is −2, and the terms inside are x and −7.
- (−2) × x = −2x.
- (−2) × (−7) = +14, because a negative times a negative is positive.
- So −2(x − 7) = −2x + 14. The sign of the second term flipped.
- Check with x = 3: −2(3 − 7) = −2 × (−4) = 8, and −2 × 3 + 14 = −6 + 14 = 8. Both match.
Answer: Answer: −2x + 14
Example 4. Expand −(4x + 9).
- A minus sign in front of parentheses means multiply by −1: −(4x + 9) = (−1)(4x + 9).
- (−1) × 4x = −4x and (−1) × 9 = −9.
- Every sign inside flips, giving −4x − 9.
- Check with x = 3: −(12 + 9) = −21, and −12 − 9 = −21. Both match.
Answer: Answer: −4x − 9
Example 5. Factor 8x + 20.
- Find the largest number that divides both terms. 8 and 20 are both divisible by 4, and nothing larger works for both.
- Divide each term by 4: 8x ÷ 4 = 2x and 20 ÷ 4 = 5.
- Write the 4 outside and what is left inside: 4(2x + 5).
- Check by distributing: 4 × 2x + 4 × 5 = 8x + 20. It matches the start.
Answer: Answer: 4(2x + 5)
Example 6. Use the distributive property to find 6 × 47 in your head.
- Split 47 into 50 − 3, which is easy to multiply: 6 × 47 = 6(50 − 3).
- Distribute: 6 × 50 − 6 × 3 = 300 − 18.
- Subtract: 300 − 18 = 282.
- Check by splitting a different way: 6(40 + 7) = 240 + 42 = 282.
Answer: Answer: 6 × 47 = 282
In plain English
Say you are packing 6 gift bags. Each bag gets 3 stickers and 2 pencils.
How many stickers do you need? 6 × 3 = 18. How many pencils? 6 × 2 = 12. In all, that is 18 + 12 = 30 things.
You can also look at one bag as “3 stickers and 2 pencils.” That is 5 things, and you have 6 bags of that: 6 × 5 = 30. The same total either way. So 6(3 + 2) = 6 × 3 + 6 × 2.
That is the distributive property. The number outside the parentheses has to reach everything inside. If you give the 6 only to the stickers, the pencils go missing.
See it
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Game night
You invite 5 friends over and ask each of them to bring some snacks and some games. Say each friend brings s snacks and g games. You can count the stuff two ways: 5 friends times “snacks and games,” or 5 groups of snacks plus 5 groups of games.
How it connects: 5(s + g) = 5s + 5g. The number of friends multiplies every item on the list, so nobody’s second item gets forgotten. The limit: it works because every friend brings the same amounts, s snacks and g games. If only one friend brought games, there would be g games, not 5g.
Honey, I Shrunk the Kids
In the movie Honey, I Shrunk the Kids, an inventor’s shrink machine accidentally shrinks some kids to a tiny size. Now imagine a much gentler ray that cuts everything it hits to half its size. Zap a 12-centimeter pencil and an 8-centimeter marker, and you get a 6-centimeter pencil and a 4-centimeter marker.
How it connects: A fraction outside the parentheses works like any other number: (1/2)(12 + 8) = (1/2)(12) + (1/2)(8) = 6 + 4 = 10. Halving the whole set gives the same result as halving each piece. The limit: every piece gets halved, not just the first one.
Measure twice, cut once
Carpenters say this to remind themselves to double-check before making a cut that cannot be undone.
How it connects: Before you trust an expansion, test it with a number. Suppose you expanded 7(x + 1) and got 7x + 1. Try x = 3: 7(3 + 1) = 28, but 7 × 3 + 1 = 22. They disagree, so something is wrong. Use a test number like 3 rather than 0, 1 or 2, because those can hide mistakes (at x = 2, x2 and 2x are both 4). One matching number is a good sign, not a proof, so if your answer has an x2, try a second number too.
Cover all your bases
To “cover all your bases” means to make sure you have handled every part of a situation so nothing catches you by surprise. The phrase is borrowed from baseball, where fielders have to cover the bases.
How it connects: In 3(x + 5) there are two bases: the x and the 5. The 3 has to reach both. Writing 3x + 5 leaves a base uncovered. The correct expansion is 3x + 15.
Points on the court
A basketball team has 5 players on the court. Say each player makes one 2-point shot and one 3-point shot. Each player scored 2 + 3 = 5 points, so the team scored 5 × 5 = 25. Or count shots instead: 5 twos and 5 threes make 10 + 15 = 25.
How it connects: 5(2 + 3) = 5 × 2 + 5 × 3, so 25 = 10 + 15. Multiplying the whole sum and multiplying each part separately give the same total.
Reversed controls
In some games, a power-up or curse reverses your controls: push the stick right and you go left, push up and you go down. Push right and up together, and you move left and down. Both parts of the move flip.
How it connects: A minus sign in front of parentheses flips every term inside: −(x + 4) = −x − 4, and −(x − 4) = −x + 4. A term that was negative becomes positive. The limit: a minus sign only flips signs. It does not change sizes, because it multiplies by −1 and nothing else.
Identical goodie bags
You have 12 cookies and 18 brownies, and you want to make identical goodie bags with nothing left over. The most bags you can make is 6, because 6 is the largest number that divides both 12 and 18. Each bag gets 2 cookies and 3 brownies.
How it connects: 12 + 18 = 6(2 + 3). Pulling out the greatest common factor, 6, is the distributive property run backward: the 6 is the number of bags and the parentheses hold one bag. (Making 3 bags of 4 cookies and 6 brownies also works, but 6 bags is the fully factored form.)
The left-out 2
Why did the 2 in 3(x + 2) feel left out? Somebody wrote 3x + 2, and the 3 never came over to say hi.
How it connects: In a(b + c) = ab + ac, the outside number multiplies every term inside. Skipping a term, like writing ab + c, is the classic mistake. For 3(x + 2) the correct answer is 3x + 6.
Memory help
DEAL with it
Draw an arrow from the outside number to Every term. Attend to the signs. Look back: test your answer with a number like 3. Say it as “DEAL with it.”
Outside in
Outside in, one by one, hit each term until you’re done. When you multiply, same signs make a plus, and different signs make a minus.
Draw the arrows
Draw an arc from the outside number to each term inside, like a rainbow with one stop for every term. Count the arcs. The number of arcs should match the number of terms. Then write each product.
Sketch the rectangle
Draw a rectangle as tall as the outside number. Split its width into the terms. Write the area of each small piece, then add them up. For 4(x + 6) the pieces are 4x and 24. (This sketch works best for positive terms, but the rule is true for all numbers.)
The skipped second term
9(y + 2) is not 9y + 2. The 9 multiplies the 2 as well, so the answer is 9y + 18. If an arrow does not land on every term, you are not finished.
A negative flips the sign inside
−8(x − 5): the second term is −5, and (−8) × (−5) = +40. So the answer is −8x + 40, not −8x − 40. Say it out loud: “negative times negative is positive.”
The matching jackets
Imagine the terms 12y and 30 are two players who both wear the same team jacket: the number 6. To factor, collect the jacket and hang it outside the parentheses. What is left of each player goes inside: 6(2y + 5). To check, hand the jacket back to everyone (distribute).
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
- Draw an arrow from the outside number to each term inside (counting each term’s sign).
- Multiply the outside number by each term, one at a time.
- Write the products, then combine any like terms.
- 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
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
- TodayRead the lesson, then try the flashcards.
- TomorrowRecall it from memory first, then check.
- In 3 daysTake the true/false and practice questions.
- In 1 weekMix it with older topics and teach it back.
- In 3 weeksA last quick review before it fades.
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