The distributive property is written in one line — a(b + c) = ab + ac — and it is usually drilled as a mechanical rule: "multiply the thing outside by each thing inside." That rule works, but it hides the geometry that makes it obvious. Once you see the property as an area being cut in two, you never forget which term gets multiplied by which, you never miss a sign, and you see why the rule extends unchanged to three terms, four terms, fractions, and negatives.
The widget below shows the geometry in motion. You set the three numbers a, b, c. The animation first draws a single rectangle of width a and height b + c, labelled with its total area a(b + c). Then a horizontal cut appears at height b, splitting the rectangle into a top sub-rectangle of area ab and a bottom sub-rectangle of area ac. Arrows fly from the original a outside the parenthesis into each piece, carrying the multiplication inside. The numeric readout verifies that both forms agree down to the last digit.
The widget
Why the picture works: the rectangle model
Multiplication of two positive numbers is an area. If you have a rectangle of width w and height h, the area is wh — that is the definition of multiplication, at the level of physical tiles. This is the oldest picture in mathematics, and it is the one the Greeks used when they proved the distributive law in Book II of Euclid's Elements.
Now imagine a rectangle of width a and total height b + c, where the height is measured as "b units first, then c more units stacked on top." There is only one rectangle. Its area is width times height, which is a(b + c).
But you can also imagine drawing a horizontal line at height b, splitting the rectangle into two stacked strips:
- A top strip of width a and height b, with area ab.
- A bottom strip of width a and height c, with area ac.
The total area has not changed — you have not added or removed any paint, only drawn a line. So
That is the distributive property. It is not a rule you need to memorise; it is the statement that a rectangle cut horizontally has the same area as the sum of its strips. Every time you expand a bracket in algebra, you are really cutting a rectangle.
What the factor a "distributes over"
The name distributive comes from the image of the factor a being handed out — distributed — to each term inside the parenthesis. Before the cut, a is the common width of a single tall rectangle. After the cut, the same a is the width of each strip. The arrows in the widget show that handing out: the a outside b + c does not disappear when the parenthesis opens; it becomes the coefficient of every term that was inside.
This is exactly why the law extends to more than two terms without any new idea. If the height is b + c + d, the rectangle splits into three strips of areas ab, ac, ad, and
Four terms, five terms, a hundred terms — the pattern is the same. You are cutting a tall rectangle into horizontal strips.
Worked examples — including negatives
Example 1. 3(4 + 2).
Set the sliders to a = 3, b = 4, c = 2 and press Play. The full rectangle has width 3 and height 6; its area is 18. After the cut, the top strip is 3 \times 4 = 12 and the bottom strip is 3 \times 2 = 6. Total: 12 + 6 = 18. Identity holds. You could have done the arithmetic without the picture, but the picture is what makes the rule memorable.
Example 2. -2(x + 5) = -2x - 10.
This is the one that trips students. Why does the minus sign reach the 5? Because the factor -2 is the common width — and the width does not change when you cut the rectangle horizontally. Whatever sign -2 has, it multiplies both strips.
Algebraically:
Why the sign carries through: distributivity is an identity, valid for all real numbers, including negatives. It is not a rule about positives that we extend by hand — it is the statement that the area of a rectangle equals the sum of its strips, and the algebra of signed numbers is designed precisely so the same identity keeps holding once lengths are allowed to be negative.
Set the widget to a = -2, b = 3, c = 5. The readout shows (-2)(3 + 5) = -16 and (-2)(3) + (-2)(5) = -6 + (-10) = -16. Identity holds. Now set a = -2, b = 1, c = 5 — closer to the symbolic case -2(x + 5) with x = 1: the numeric expansion matches symbolic expansion perfectly.
Example 3. 5(a - 3).
Rewrite the subtraction as addition of a negative: a - 3 = a + (-3). Then distribute:
The minus sign inside the parenthesis is just a negative number in disguise. Distribute to every term, keeping each term's sign.
Example 4. (x + 2)(x + 3) — distribute twice.
Treat (x + 2) as the factor a and distribute it over x + 3:
Now distribute again, this time with x (then 3) as the outer factor:
Every FOIL expansion, every quadratic product, every binomial identity like (a + b)^2 = a^2 + 2ab + b^2 — all of them are the distributive property applied more than once. There is only one multiplication rule in algebra, and you just watched it.
Where to go next
For the algebraic catalogue of identities that ride on top of distributivity — (a + b)^2, (a - b)^2, (a + b)(a - b), (a + b + c)^2, and the related factorisation patterns — continue to Algebraic Identities. For the vocabulary of terms, coefficients, and degrees that lets you talk about expressions precisely, and for the rules of combining like terms, see the parent article on Algebraic Expressions.