The Distributive Property

Mental arithmetic works well on small equations, but it does not scale. Algebra offers something better: properties — rules that are always true and can be applied methodically. The distributive property is the one you will reach for most often.

Concept What an equals sign claims

Writing $a = b$ is a claim that both sides have exactly the same value. Two expressions that always match in this way are called equivalent.

5 + 3 = 8

— the left side works out to 8, and so does the right.

This matters because every algebraic property is really a statement that one expression may be swapped for an equivalent one without changing anything.

Concept The distributive property

a(b + c) = ab + ac

A multiplier sitting outside a bracket reaches every term inside it. Multiply each term separately, then add the results.

The property applies whenever a multiplication meets a bracket containing two or more terms, and it works no matter how many terms there are.

Concept Why it holds

You can always check the property against the order of operations, which says to clear the bracket first. Both routes must agree.

Bracket first:

3(4 + 6) = 3 \times 10 = 30

Distributing:

3 \times 4 + 3 \times 6 = 12 + 18 = 30

They agree, as they must. This also gives you a free way to verify any distribution you are unsure about.

Example Three distributions

3(4 + 6) = 12 + 18 = 30

6(1 + 4) = 6 + 24 = 30

7(5 + 2) = 35 + 14 = 49

In each case the outside number multiplies both terms, never just the first one. Missing the second term is the single most common error.

Example Mental arithmetic made easier

Work out $5 \times 27$ without writing anything down.

Split 27 into friendlier parts:

27 = 20 + 7

Distribute:

5(20 + 7) = 5 \times 20 + 5 \times 7

= 100 + 35

⟹ 135

Splitting a number into a round part and a remainder turns an awkward multiplication into two easy ones. This is the property doing genuine work, not just theory.

Example Distributing over a variable

Expand $4(x + 3)$ .

4 \times x = 4x

4 \times 3 = 12

⟹ 4x + 12

Here the bracket cannot be cleared first, because $x$ is unknown — so distribution is the only way forward. That is precisely why the property is indispensable in algebra.

Note Mistakes to avoid

Multiplying only the first term:

3(4 + 6)

is not

12 + 6

Adding the outside number instead of multiplying by it.

Dropping the bracket without distributing at all.

Forgetting that the property extends to three or more terms inside.

Trying to clear a bracket that contains an unknown instead of distributing.

Summary

An equals sign states that both sides have the same value.
The distributive property says a(b + c) = ab + ac.
The multiplier must reach every term inside the bracket.
Clearing the bracket first gives the same answer, which is a useful check.
When the bracket contains an unknown, distributing is the only option.