What is the distributive property?

It says that multiplying a sum gives the same answer as multiplying each part and then adding: a(b + c) = ab + ac. It is what lets you remove brackets in algebra, and it is the reason you can multiply 7 × 23 in your head as 7 × 20 plus 7 × 3.

Why is a(b + c) = ab + ac true?

Because both sides are the area of the same rectangle. A rectangle a tall and (b + c) wide has area a(b + c). Cut it with a vertical line into a piece b wide and a piece c wide and the two areas are ab and ac. Cutting a shape does not change how much of it there is, so the two expressions must be equal.

How does it work with a negative sign?

The sign travels onto every term inside the brackets. So -2(x + 3) is -2x - 6, not -2x + 6: both terms get multiplied by the -2. This is the single most common mistake with brackets, and it is worth checking every time by asking what happens when you substitute a number.

What is the difference between distributing and factoring?

They are the same fact used in opposite directions. Distributing goes from a(b + c) to ab + ac, removing brackets. Factoring goes from ab + ac back to a(b + c), putting them in. Being able to run the rule backwards is what makes factoring possible at all.

Is FOIL the distributive property?

Yes — FOIL is just the distributive property applied twice. To expand (x + 2)(x + 3) you distribute the whole first bracket over the second, then distribute again inside each part. FOIL is a memory aid for the four products that result; it is not a separate rule, which is why it stops working once a bracket has three terms and distributing still does.

The Distributive Property

a(b + c) = ab + ac is usually stated and then drilled. It is easier to remember once you have seen it, because it is a picture: one rectangle, cut in two.

By Nethanel Bar, Co-founder & CEO

Last updated August 25, 2026

You have almost certainly used the distributive property without being told its name. Asked for 7 × 23, most people do 7 × 20 = 140, then 7 × 3 = 21, then add to get 161. That is 7(20 + 3) = 7*20 + 7*3 — the rule, applied to arithmetic.

In algebra it does the same job on letters, and it is the reason brackets can be removed at all. Here is why it is true.

3(4 + 2) = 3*4 + 3*2 = 18

Cutting the rectangle does not change its area. That is the whole of the distributive property.

It is one rectangle, cut

Set the sliders to a = 3, b = 4, c = 2 and look at the shape.

The whole rectangle is 3 tall and 4 + 2 = 6 wide, so its area is 3(4 + 2) = 18. The vertical cut splits it into a 3-by-4 piece and a 3-by-2 piece, with areas 3*4 = 12 and 3*2 = 6 — and 12 + 6 is 18.

That is the entire proof. Cutting a rectangle does not change its area, so the one expression and the two-part expression must be equal, whatever the numbers are. Move all three sliders and the two totals never disagree.

Written in general: a(b + c) = a*b + a*c.

The width and height labels are worth noticing, because they say which number is doing what. The a is outside the brackets and it multiplies both pieces, since both pieces are a tall. Nothing inside the brackets multiplies anything else.

The negative case, which is where people slip

Now drag c below zero.

The second rectangle changes colour and moves inside the first, because a negative width means the piece is being taken away. With a = 3, b = 4 and c = -2, the total is 3(4 - 2) = 6 — and the two pieces are 3*4 = 12 and minus 3*2 = 6. Twelve take away six.

So the minus sign travels onto the second term:

a(b - c) = a*b - a*c

This is the mistake worth guarding against. Writing -2(x + 3) as -2x + 6 is wrong; the -2 multiplies both terms, giving -2x - 6. If you are ever unsure, substitute a number: at x = 1 the original is -2(1 + 3) = -8, and only -2x - 6 gives -8.

Using it to solve something

Removing brackets is rarely the point in itself — it is a step towards getting a letter on its own. Here it is inside a solve:

Worth noticing: the route above did not expand the brackets at all, because with 27 divisible by 3 it was quicker to divide both sides first. Distributing is a tool, not an obligation — expanding to 3x + 12 = 27 reaches the same x = 5 by a longer path.

Running it backwards

Read a*b + a*c = a(b + c) from right to left and you have factoring: spotting a factor shared by every term and pulling it out front. 6x + 9 becomes 3(2x + 3) because both terms are divisible by 3.

It is the same picture, looked at the other way: instead of cutting a rectangle you are noticing that two rectangles have the same height and can be pushed together.

Try one

That one has the negative on the outside on purpose. If you get -1, the minus sign did not reach the 3.

Common questions

What is the distributive property?: It says that multiplying a sum gives the same answer as multiplying each part and then adding: a(b + c) = ab + ac. It is what lets you remove brackets in algebra, and it is the reason you can multiply 7 × 23 in your head as 7 × 20 plus 7 × 3.
Why is a(b + c) = ab + ac true?: Because both sides are the area of the same rectangle. A rectangle a tall and (b + c) wide has area a(b + c). Cut it with a vertical line into a piece b wide and a piece c wide and the two areas are ab and ac. Cutting a shape does not change how much of it there is, so the two expressions must be equal.
How does it work with a negative sign?: The sign travels onto every term inside the brackets. So -2(x + 3) is -2x - 6, not -2x + 6: both terms get multiplied by the -2. This is the single most common mistake with brackets, and it is worth checking every time by asking what happens when you substitute a number.
What is the difference between distributing and factoring?: They are the same fact used in opposite directions. Distributing goes from a(b + c) to ab + ac, removing brackets. Factoring goes from ab + ac back to a(b + c), putting them in. Being able to run the rule backwards is what makes factoring possible at all.
Is FOIL the distributive property?: Yes — FOIL is just the distributive property applied twice. To expand (x + 2)(x + 3) you distribute the whole first bracket over the second, then distribute again inside each part. FOIL is a memory aid for the four products that result; it is not a separate rule, which is why it stops working once a bracket has three terms and distributing still does.

Now do it yourself, properly

Reading someone else's working is not the same as being able to do it. The Coddy math course puts you on a board that checks every move you make, so you find out where you actually stand.

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