Algebra
Expressions
The distributive property
20 practice questions
2 video lessons
Theory + worked examples
Theory
The distributive property multiplies a factor across each term in parentheses:
\[a(b+c)=ab+ac,\qquad a(b-c)=ab-ac.\]
Reversed, it factors out a common factor.
Distribute to every term, keeping track of signs.
The area model: \(a(b+c)=ab+ac\).
The distributive property.
The property:
\[a(b+c)=ab+ac\]
A negative outside changes the sign of every inside term.
How to distribute
- Multiply the outside factor by the first term.
- Multiply it by each remaining term.
- Keep the signs correct.
- To factor, pull out the greatest common factor.
Example 1 — Distribute over a sum
Expand \(3(x+4)\).
Solution
Multiply \(3\) by each term.
| \(3(x+4)\) | \(=\) | \(3x+12\) |
Example 2 — Distribute over a difference
Expand \(5(2x-3)\).
Solution
Distribute to both terms.
| \(5(2x-3)\) | \(=\) | \(10x-15\) |
Example 3 — Negative multiplier
Expand \(-2(x-5)\).
Solution
Watch the signs.
| \(-2(x-5)\) | \(=\) | \(-2x+10\) |
Example 4 — Factor out (reverse)
Factor \(6x+9\) using the distributive property.
Solution
Pull out the common factor \(3\).
| \(6x+9\) | \(=\) | \(3(2x+3)\) |
Common pitfalls
Distribute to every term, not just the first.
A negative multiplier flips every sign.
Factoring is the reverse of distributing.
Frequently asked questions
What is the distributive property?
\(a(b+c)=ab+ac\) — multiply the factor across each term.
How do you expand \(-3(x-2)\)?
\(-3x+6\) — distribute the negative to both terms.
How is factoring related?
Factoring reverses the distributive property by pulling out a common factor.
Do you distribute to every term?
Yes — every term inside the parentheses.
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