Distributive Property Calculator
Enter the three values in your expression and choose whether to distribute over numbers or variables. The calculator expands a(b + c) into ab + ac instantly, shows every step of the working, and gives you the simplified result. Switch between a numeric mode for arithmetic and an algebraic mode for expressions with x.
What is the distributive property?
The distributive property of multiplication states that multiplying a factor by a sum (or difference) gives the same result as multiplying the factor separately by each addend and then adding the products. Formally: a(b + c) = ab + ac. This identity works for any real numbers a, b, and c, including negatives, fractions and decimals. It is one of the three core field axioms (alongside commutativity and associativity) that underpin all of arithmetic and algebra. The name comes from the way a is "distributed" across each term inside the parentheses.
How to use this calculator
Choose a mode: "Numeric" for expressions where all three values (a, b, c) are plain numbers, or "Algebraic" for expressions of the form a(bx + c) where b is the coefficient of x. Select whether the inner operation is addition or subtraction. Enter your values for a, b, and c, and the calculator immediately shows the expanded form, each individual product, and the simplified result. The "Show your work" panel walks through every arithmetic step in the same order a student would write it. Numeric mode also verifies the answer by computing the bracket first, so you can see that both paths reach the same number.
Numeric example: 3(4 + 2)
Start with 3(4 + 2). Multiply 3 by 4 to get 12, then multiply 3 by 2 to get 6. Adding the two products: 12 + 6 = 18. To check: 4 + 2 = 6, and 3 x 6 = 18. Both methods agree. For subtraction, 3(4 - 2) works the same way: 3 x 4 = 12, 3 x 2 = 6, and 12 - 6 = 6. Notice that the bracket shortcut (4 - 2 = 2, times 3 = 6) matches as well.
Algebraic example: 3(2x + 5)
In algebraic mode the calculator handles expressions like 3(2x + 5). Distribute 3 over 2x: 3 x 2x = 6x. Distribute 3 over 5: 3 x 5 = 15. The expanded form is 6x + 15. Unlike numeric mode there is no single number answer because x is unknown, but the expression is now in its standard expanded form ready for further algebra. With negative coefficients, for example -2(3x - 4), the same rule applies: -2 x 3x = -6x and -2 x (-4) = 8, giving -6x + 8. The step-by-step panel makes every sign change explicit.
Common mistakes and how to avoid them
The most frequent error is forgetting to multiply the outside factor by every term inside the brackets, distributing only to the first term. For example, writing 3(x + 5) = 3x + 5 instead of 3x + 15. A second common error involves signs: with a negative outside factor such as -4(x - 3), forgetting that -4 x (-3) = +12, not -12. The "show your work" panel in this calculator labels each multiplication individually so neither term is missed and sign changes are explicit. A third mistake is trying to combine unlike terms in algebraic mode: 6x + 15 cannot be simplified further because 6x and 15 do not share the same variable factor.
The distributive property in factoring
Factoring is the distributive property run in reverse: ab + ac is rewritten as a(b + c) by pulling out the common factor a. For example, 6x + 9 shares a factor of 3, so it becomes 3(2x + 3). This reverse application is the basis of factoring polynomials, simplifying fractions, and solving equations. The FOIL method for two binomials, (a + b)(c + d) = ac + ad + bc + bd, is just the distributive property applied twice: first distribute (a + b) over c, then over d.
Distributive property rules and identities
| Form | Expression | Example |
|---|---|---|
| Addition (numeric) | a(b + c) = ab + ac | 4(3 + 2) = 12 + 8 = 20 |
| Subtraction (numeric) | a(b - c) = ab - ac | 5(7 - 3) = 35 - 15 = 20 |
| Negative multiplier | -a(b + c) = -ab - ac | -2(4 + 1) = -8 - 2 = -10 |
| Variable term | a(bx + c) = abx + ac | 3(2x + 5) = 6x + 15 |
| Two binomials (FOIL) | (a + b)(c + d) = ac + ad + bc + bd | (x + 2)(x + 3) = x^2 + 5x + 6 |
| Factoring (reverse) | ab + ac = a(b + c) | 6x + 9 = 3(2x + 3) |
| Over subtraction (algebra) | a(bx - c) = abx - ac | 4(3x - 2) = 12x - 8 |
| Distributive over 3 terms | a(b + c + d) = ab + ac + ad | 2(1 + 3 + 5) = 2 + 6 + 10 = 18 |
Key forms of the distributive property used in arithmetic and algebra.
Frequently asked questions
What is the distributive property formula?
The formula is a(b + c) = ab + ac. The factor a outside the parentheses is multiplied by every term inside. For subtraction the rule is a(b - c) = ab - ac. Both forms hold for any real numbers, including negatives, fractions and decimals.
Does the distributive property work with subtraction?
Yes. The distributive property of multiplication over subtraction states that a(b - c) = ab - ac. The outside factor is distributed to both terms and the subtraction sign belongs to the second product. For example, 5(7 - 3) = 5 x 7 - 5 x 3 = 35 - 15 = 20, which equals 5 x 4 = 20 computed the short way.
How does the distributive property work with negative numbers?
A negative outside factor changes the sign of every term it distributes to. For example, -3(2 + 5) = -3 x 2 + (-3) x 5 = -6 - 15 = -21. With subtraction inside: -4(x - 3) = -4x + 12, because -4 multiplied by -3 is positive 12. Keeping track of each sign separately when you distribute is the key to avoiding errors.
Can the distributive property be used with variables?
Yes, and this is one of its most important uses. When a, b, or c involve variables, the same rule applies: a(bx + c) = abx + ac. Unlike in numeric mode you cannot add the two terms together unless they are "like terms" (same variable and exponent), so the result stays as an expanded polynomial expression.
What is the difference between expanding and factoring?
Expanding means applying the distributive property to remove parentheses: a(b + c) becomes ab + ac. Factoring is the reverse: finding a common factor to reintroduce parentheses, so ab + ac becomes a(b + c). Both steps use the same property; they just run it in opposite directions.
What is the FOIL method and how does it relate?
FOIL (First, Outer, Inner, Last) is a mnemonic for multiplying two binomials: (a + b)(c + d) = ac + ad + bc + bd. It is the distributive property applied twice: first distribute (a + b) over c to get ac + bc, then distribute (a + b) over d to get ad + bd, and add everything together. So FOIL is not a separate rule, it is a consequence of the distributive property.
Why is the distributive property important in arithmetic?
It lets you split a multiplication into easier pieces. For instance, 7 x 98 is tricky to compute directly, but distributing gives 7 x (100 - 2) = 700 - 14 = 686, which is far easier mentally. This strategy, sometimes called "partial products" or "break-apart," is one of the core strategies taught in elementary and middle school math.