What is the distributive property?
Type a product with brackets, such as 3(x + 4) or (2x + 1)(3x² − x + 4). The distributive property calculator multiplies each term of one factor by each term of the other, shows every product, and combines like terms.
- Expanded
- 3x + 12
3(x + 4) = 3x + 12.
- Each product
- 3x + 12
- Steps
- 3 × (x + 4) = 3 × x = 3x, 3 × 4 = 12; Add: 3x + 12
Expanded: 3x + 12. 3(x + 4) = 3x + 12.
How is each term multiplied?
How to calculate
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.
Example with the default inputs (Product 3(x + 4)): 3(x + 4) = 3x + 12.
Method: a(b + c) = ab + ac: every term of one factor times every term of the other, then like terms combined.
- 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
Each example is checked against the calculator on every build.
- Product 3(x + 4) gives Expanded 3x + 12, Each product 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
- Product (2x + 1)(3x^2 - x + 4) gives Expanded 6x^3 + x^2 + 7x + 4, Each product 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
- Product -2x(3x - 5) gives Expanded -6x^2 + 10x, Steps (−2x) × (3x − 5) = (−2x) × 3x = −6x², (−2x) × (−5) = 10x; Add: −6x² + 10x.
- Product 6(10 + 3) gives Expanded 78, Each product 60 + 18.
- Product (x + 4)0.5 gives Expanded x/2 + 2.
How it works
The distributive property: a(b + c) = ab + ac, and in general every term of one factor is multiplied by every term of the other.
Input. One product of two factors, written side by side or with *, such as 3(x + 4), (x + 4)3, −2x(3x − 5) or (x − 1)(x^2 + x + 1). At least one factor must be a sum of two or more terms as typed: 10 + 3 stays two terms, and a term that is 0 is left out. A term that is itself a product with a bracket of two or more terms, as in 3(x + 4)(x + 1) or 2(x + 3(y + 1)), gets a message: multiply out one product at a time. A leading minus sign belongs to the first factor: −2x(3x − 5) is (−2x)(3x − 5). Terms may hold numbers, letters a to z except e, whole powers and products. A sum on its own, or a product of three factors, gets a message instead of an answer.
Order. The factor with fewer terms is the outer factor (the first factor when both have the same number of terms). The steps take each term of the outer factor in the order typed and multiply it by each term of the inner factor in the order typed.
Exact arithmetic. Every coefficient is an exact fraction: 0.5 is 1/2. Each product is exact, and like terms (the same letters to the same powers) are added exactly.
Output format. "Each product" lists the products in the order above, joined with + and −. "Expanded" is the sum with like terms combined, highest total power first, then letters in alphabetical order, then the number. When no like terms combine, the steps end at the sum. A factor of 0 gives 0, in one step: 0 × (x + 1) = 0. In the steps a negative, fractional or multi-factor term is put in brackets, powers show as superscripts, and minus signs are true minus signs.
Worked examples by hand
3(x + 4) (OpenStax Prealgebra Example 7.17). 3 × x = 3x and 3 × 4 = 12: 3x + 12.
(2x + 1)(3x² − x + 4) (OpenStax Example 4). 2x times each term: 6x³, −2x², 8x. 1 times each term: 3x², −x, 4. Sum: 6x³ − 2x² + 8x + 3x² − x + 4. Combine like terms: 6x³ + x² + 7x + 4.
−2x(3x − 5). (−2x)(3x) = −6x² and (−2x)(−5) = 10x: −6x² + 10x.
6(10 + 3). 60 + 18 = 78, which is 6 × 13.
(x + 4)0.5. The single factor 1/2 goes first: x/2 + 2.
Other questions people ask
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.