Matemáticas
The Distributive Property
a(b + c) = ab + ac, drawn as a rectangle split in two. The same rule does 7 × 34 in your head and turns 3(2x − 5) into 6x − 15 — plus what it does not allow.
The distributive property
The distributive property states that multiplying a sum gives the same result as multiplying each part separately and adding, so a(b + c) = ab + ac.
- También llamado
- Distributive law, Expanding brackets, Multiplying out
- Dónde lo encuentra el alumnado
- Grade 6 mathematics as a mental-arithmetic idea and inside the grid method, then formally in Grade 7 or 8 when brackets are first expanded in algebra.
La respuesta corta
The distributive property says that a(b + c) = ab + ac: multiplying a bracket is the same as multiplying each term inside it and adding the results. It is what lets 7 × 34 be done as 7 × 30 + 7 × 4 = 238, and what turns 3(2x − 5) into 6x − 15.
Un ejemplo
7 × 34 = 7 × 30 + 7 × 4 = 210 + 28 = 238
34 is split into 30 + 4, each part is multiplied by 7, and the results are added. The identical move in algebra is 3(2x − 5) = 6x − 15: multiply the 2x by 3 and the −5 by 3. Arithmetic and algebra are not two rules here; they are one rule with different contents.
The rectangle that makes it obvious
Draw a rectangle 7 tall and 34 wide, then cut it with a vertical line so the width splits into 30 and 4. You now have two rectangles, 7 × 30 and 7 × 4, and their areas add to the area of the original. Nothing was added or removed by the cut.
That picture is the whole property. a(b + c) is the area of one rectangle; ab + ac is the area of the two pieces. Because the areas must be equal, the expressions must be equal, for every value of a, b and c.
One rule, several disguises
Most of the multiplication a student does is this property wearing different clothes. The grid method for long multiplication is a rectangle cut into pieces by place value. Mental arithmetic uses it in the other direction: 6 × 99 is easier as 6 × 100 − 6 × 1 = 594, distributing over a subtraction.
Read backwards, it becomes factorising. If 3(2x − 5) = 6x − 15, then 6x − 15 = 3(2x − 5), and spotting the common factor is the same knowledge used in reverse. Students who see expanding and factorising as two unrelated topics are usually missing that they are one equation read in two directions.
- 7 × 34 = 7 × 30 + 7 × 4
- 6 × 99 = 6 × 100 − 6 × 1
- 3(2x − 5) = 6x − 15
- 6x − 15 = 3(2x − 5), the same statement reversed
What it does not license
The property distributes multiplication over addition and subtraction, and nothing else. It does not distribute over another multiplication: a(bc) is not ab × ac, because you would be multiplying by a twice. Nor does a power distribute over a sum, which is why (a + b)² is not a² + b² — expand it properly and the 2ab in the middle is the term being lost.
The other place it goes wrong is signs. In −3(2x − 5), the multiplier is −3, so both terms change sign: −6x + 15. Writing −6x − 15 is by far the most common expanding error, and it comes from multiplying the first term by −3 and the second by 3.
Preguntas frecuentes
Is expanding brackets the same as the distributive property?
Yes — expanding is the property in action, and 'distributive property' is the name of the rule that makes expanding valid. UK schools tend to say 'expand' or 'multiply out'; American textbooks and the SAT tend to name the property. Both refer to a(b + c) = ab + ac.
Does it work with subtraction?
Yes. a(b − c) = ab − ac, because subtracting is adding a negative. So 5(x − 4) = 5x − 20 and 8 × 97 = 8 × 100 − 8 × 3 = 776. The only care needed is the sign: whatever is in front of the bracket multiplies the sign of each term as well as its size.
Why does −(x − 4) become −x + 4?
Because the minus sign in front is a multiplier of −1, and it distributes over both terms. −1 × x is −x, and −1 × −4 is +4. Treating the bracket as though only the first term is affected gives −x − 4, which is the standard slip when a negative sits outside brackets.
Does the distributive property apply to division?
It applies when you divide a sum, not when you divide by one. (a + b)/c does split into a/c + b/c, so (6x + 9)/3 = 2x + 3. But c/(a + b) does not split: 12/(2 + 4) is 2, while 12/2 + 12/4 is 9. Dividing by a bracket is not distribution.
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