The distributive property
Expanding a bracket term by term, the same rule read backwards as factoring, and one expression simplified all the way down with each step named.
Reference chart
The distributive property
Multiply over a sum
- Over a sumEvery term takes it.3(x + 4) = 3x + 12
- Over a differenceThe minus travels too.5(x − 2) = 5x − 10
- A negative outsideBoth signs flip.−2(x − 5) = −2x + 10
Simplify 3(x + 4) − 2x + 7 — naming each step
- 1Distributive3x + 12 − 2x + 7
- 2Commutative — reorder3x − 2x + 12 + 7
- 3Combinex + 19
Backwards, it is factoring
- Find what divides bothTake it outside.6x + 15 = 3(2x + 5)
- Check itMultiply back out.3(2x + 5) = 6x + 15
Not this — this
- 3(x + 4)is not 3x + 4.the 4 gets multiplied too
- −2(x − 5)is not −2x − 10.minus times minus is plus
- 3(x + 4)is not 3x × 3(4).multiply once, into each term
What's on this chart
The distributive property is the one that joins the two operations: a number outside a bracket multiplies everything inside it, not just the first thing. That is the whole rule, and 3(x + 4) = 3x + 4 is the most common error in a first algebra course — which is why it is on this chart twice, once as what to do and once as what not to.
Why this chart helps
Signs are where it goes wrong after that. The minus in 5(x − 2) travels, and a negative outside flips both signs: −2(x − 5) is −2x + 10, not −2x − 10. Both are on the card with the working, because a rule about signs stated without an example is a rule a reader will apply confidently in the wrong direction.
More printable charts
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