Distributive Property
a(b+c) = ab + ac — multiply the factor by each term inside the parentheses. Works with subtraction and multi-term. Essential for mental math, algebra, and factoring.
Why This Mathematical Concept Matters
Why: The distributive property underlies multiplication of sums and is fundamental to algebra.
How: Multiply the factor a by each term inside: a×b + a×c = ab + ac. For subtraction: a(b−c) = ab − ac.
- ●Mental math: 7×98 = 7(100−2) = 700 − 14 = 686.
- ●Area model: rectangle width a, length (b+c) has area ab + ac.
- ●Factoring reverses it: ab + ac = a(b + c).
📐 Examples — Click to Load
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📐 Step-by-Step Breakdown
⚠️For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
The area model: a(b+c) is the area of a rectangle split into ab and ac.
— Geometry
Mental math: 23×12 = 23(10+2) = 230 + 46 = 276.
— Arithmetic
📋 Key Takeaways
- • a(b + c) = ab + ac — multiply each term inside by the factor outside
- • a(b − c) = ab − ac — works with subtraction too
- • Mental math: 7×98 = 7(100−2) = 700 − 14 = 686
- • Area model: rectangle width a, length (b+c) has area ab + ac
💡 Did You Know?
📖 How It Works
To expand a(b + c), multiply the factor a by each term inside the parentheses: a×b + a×c = ab + ac. For a(b−c), we get ab − ac. The distributive property is fundamental in arithmetic and algebra.
📝 Worked Example: 3(4+5)
Left: 3(4+5) = 3×9 = 27
Right: 3×4 + 3×5 = 12 + 15 = 27
Result: Both equal 27 ✓
🚀 Real-World Applications
🧮 Mental Math
7×98 = 7(100-2) = 700 - 14 = 686
📐 Area Model
Rectangle area: a(b+c) = ab + ac
📊 Algebra
Expand and factor expressions
💰 Finance
Distribute costs across items
🔬 Physics
Force distribution, work
📏 Measurement
Scale factors, conversions
⚠️ Common Mistakes to Avoid
- Watch signs: −2(3−5) = −6 + 10 = 4, not −6 − 10
- Distribute to ALL terms: a(b+c+d) = ab + ac + ad
- Factoring is inverse: ab + ac = a(b+c), not a(bc)
🎯 Expert Tips
💡 Watch signs
−2(3−5) = −6 + 10 = 4
💡 Mental math
Break numbers: 7×98 = 7(100−2)
💡 Verify
Substitute values to check
💡 Factoring
Reverse: ab + ac = a(b + c)
📊 Reference Table
| Form | Expansion |
|---|---|
| a(b + c) | ab + ac |
| a(b − c) | ab − ac |
| −a(b + c) | −ab − ac |
❓ FAQ
What is the distributive property?
a(b+c) = ab + ac. A number times a sum equals the sum of that number times each term.
Does it work with subtraction?
Yes. a(b−c) = ab − ac.
How do I use it for mental math?
Break numbers: 7×98 = 7(100−2) = 700 − 14 = 686.
What about three terms?
a(b+c+d) = ab + ac + ad.
What is factoring?
The reverse: ab + ac = a(b + c). Factor out the common factor a.
📌 Summary
The distributive property: a(b+c) = ab + ac. Multiply the factor by each term inside. Works with subtraction and multi-term. Essential for mental math, algebra, and factoring.
🔗 Next Steps
Try the Associative Property Calculator or the FOIL Method Calculator for binomial expansion.
⚠️ Disclaimer: Results are for educational purposes. Verify critical calculations independently.