Distributive Property Calculator (expand a(b + c) step by step)

Understanding the distributive property calculator — concept and principle

This calculator expands an expression of the form a(b + c) with the distributive property, step by step: it multiplies the number outside the brackets into every term inside, combines like terms and, when there are no letters, gives the numeric value. Use it for pre-algebra homework or to check an expansion.

The outside factor reaches every term in the brackets, not just the first.

How it is calculated

a(b + c) = ab + ac; a(b − c) = ab − ac

Every term inside the brackets is multiplied, including its sign: −2(3y − 5) = −6y + 10.
Letters keep their exponents (type powers with ^, such as x^2).
After distributing, like terms are combined; with only numbers, the value is shown.

The principle

Multiplication spreads over addition. Three bags each holding 2 apples and 5 oranges contain 3 × 2 apples and 3 × 5 oranges, the same as 3 × (2 + 5) pieces of fruit in total. As an area model, a rectangle a wide and (b + c) long splits into two rectangles of area ab and ac. The property also works backwards for mental arithmetic: 3 × 9.25 = 3 × 9 + 3 × 0.25 = 27.75, and it is the reverse of factoring out a common factor.

Worked example

Default inputs: number outside 3, expression inside 2x + 5.

Problem: 3(2x + 5).
Distribute to each term: 3·2x + 3·5.
Multiply: 3·2x = 6x and 3·5 = 15.
Like terms combined: 6x + 15, with 2 terms.
The result has a letter, so no single numeric value is shown (—).

How to read the result

Check by substituting a number: at x = 1, 3(2 + 5) = 21 and 6 + 15 = 21. If the two forms disagree for any value of x, a term was missed or a sign was dropped. The expanded form is useful for adding to other expressions; the bracketed form is often better for seeing structure.

Common mistakes

Multiplying only the first term: 3(2x + 5) ≠ 6x + 5.
Losing a negative sign: −(x − 4) = −x + 4.
Adding instead of multiplying the outside number.
Combining unlike terms such as 6x and 15.

Keep in mind: Type the expression with x as the variable and explicit operators; check that the problem line matches what you meant.

Frequently asked questions

What is the distributive property?

a(b + c) = ab + ac: multiplying a sum equals multiplying each term and adding.

How do I distribute a negative number?

Multiply each term by it, which changes every sign inside: −2(3y − 5) = −6y + 10.

Is factoring the opposite of distributing?

Yes, factoring takes a common factor back out: 6x + 15 = 3(2x + 5).

Sources: OpenStax, Prealgebra 2e (2020), §7.3 Distributive Property

Checked 9 Nov 2026. OpenStax, Prealgebra 2e (2020), §7.3 Distributive Property (if a, b, c are real numbers then a(b + c) = ab + ac; also a(b − c) = ab − ac and (b + c)a = ba + ca; worked example 3(9.25) = 3(9 + 0.25) = 27.75).

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