# What is the distributive property?

Multiplies out a product such as 3(x + 4) or −2x(3x − 5) by the distributive property, a(b + c) = ab + ac, with each product shown and like terms combined in exact fractions.

- Page: https://www.acalculator.org/math/distributive-property-calculator
- JSON spec: https://www.acalculator.org/math/distributive-property-calculator.json
- Version: 2c45b9bdbdf9

## Default answer

Example with the default inputs (Product 3(x + 4)): 3(x + 4) = 3x + 12.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| e | Product | A factor times a bracket, such as 3(x + 4), −2x(3x − 5 + y) or (x − 1)(x² + x + 1). |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| result | Expanded | The product multiplied out, with like terms combined. |
| distributed | Each product | Every term of one factor times every term of the other, before like terms are combined. |
| steps | Steps | Each product, the sum, and like terms combined. |

## Method

a(b + c) = ab + ac: every term of one factor times every term of the other, then like terms combined.

## Assumptions

- The input is one product of two factors, at least one of them a sum. Coefficients are exact fractions: 0.5 is 1/2.

## Worked examples

1. e = 3(x + 4) gives result = 3x + 12, distributed = 3x + 12, steps = 3 × (x + 4) = 3 × x = 3x, 3 × 4 = 12; Add: 3x + 12. Source: OpenStax, Prealgebra 2e, §7.3 Distributive Property, Example 7.17 (3(x + 4) = 3x + 12), https://openstax.org/books/prealgebra-2e/pages/7-3-distributive-property.
2. e = (2x + 1)(3x^2 - x + 4) gives result = 6x^3 + x^2 + 7x + 4, distributed = 6x^3 - 2x^2 + 8x + 3x^2 - x + 4. Source: OpenStax, Algebra and Trigonometry 2e, §1.4 Polynomials, Example 4 ((2x + 1)(3x² − x + 4) = 6x³ + x² + 7x + 4), https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-4-polynomials.
3. e = -2x(3x - 5) gives result = -6x^2 + 10x, steps = (−2x) × (3x − 5) = (−2x) × 3x = −6x², (−2x) × (−5) = 10x; Add: −6x² + 10x.
4. e = 6(10 + 3) gives result = 78, distributed = 60 + 18.
5. e = (x + 4)0.5 gives result = x/2 + 2.

## FAQ

### What is the distributive property?

For any numbers a, b and c, a(b + c) = ab + ac. Multiplying a sum by a number gives the same result as multiplying each term by that number and adding.

### How do I distribute a negative number?

Multiply each term by the negative number, sign included. −2x(3x − 5) = (−2x)(3x) + (−2x)(−5) = −6x² + 10x. A negative times a negative gives a positive.

### Does the distributive property work with subtraction?

Yes. a(b − c) = ab − ac, because b − c is b + (−c). For example 2(x − 3) = 2x − 6.

### How do I multiply two brackets with the distributive property?

Multiply every term of the first bracket by every term of the second, then add. (2x + 1)(3x² − x + 4) = 6x³ − 2x² + 8x + 3x² − x + 4 = 6x³ + x² + 7x + 4. For two binomials this is the FOIL method.

### Can I use it for mental arithmetic?

Yes. 6 × 13 = 6(10 + 3) = 60 + 18 = 78. Splitting a number into tens and ones makes the products easy.

### Is (x + 4)3 the same as 3(x + 4)?

Yes. Multiplication is commutative, so the single factor can be on either side. The calculator always writes the single factor first in the steps.

## Sources

- OpenStax, Prealgebra 2e, §7.3 Distributive Property (a(b + c) = ab + ac; Example 7.17: 3(x + 4) = 3x + 12). https://openstax.org/books/prealgebra-2e/pages/7-3-distributive-property (retrieved 2026-10-05)
- OpenStax, Algebra and Trigonometry 2e, §1.4 Polynomials (multiply each term of the first polynomial by each term of the second; Example 4: (2x + 1)(3x² − x + 4) = 6x³ + x² + 7x + 4). https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-4-polynomials (retrieved 2026-10-05)