The distributive property says that multiplying a number by a sum gives the same result as multiplying it by each term and then adding: a(b + c) = ab + ac. It multiplies a factor by each term inside parentheses, so 3(x + 4) = 3x + 12. It works with subtraction too, and using it backward is factoring.
In Spanish
propiedad distributiva
Also called
distribute, distributive law, distributive property of multiplication over addition
A rectangle split into two parts has the same area as the two parts added together. That is the distributive property: the outside factor multiplies every term inside.
Formal definition and notation
For all real numbersa, b and c: a(b + c) = ab + ac and a(b − c) = ab − ac. The factor may also be on the right: (b + c)a = ba + ca. We say multiplicationdistributes over addition and subtraction. Read from right to left, the property pulls out a common factor: ab + ac = a(b + c), which is factoring.
Symbol
Read it as
Meaning
a(b + c)
a times the quantity b plus c
the parentheses group the sum; the factor a multiplies all of it
−(x + 7)
the opposite of the quantity x plus 7
a hidden factor of −1: −(x + 7) = −x − 7
Key facts
Multiply the outside factor by every term inside the parentheses, not just the first one.
A negative factor changes every sign inside: −3(x − 4) = −3x + 12.
It works for any number of terms: 2(x + y − 5) = 2x + 2y − 10.
Factoring is the distributive property in reverse: 36 + 8 = 4(9 + 2).
Multiplying two binomials uses it twice: (x + 2)(x + 3) = x(x + 3) + 2(x + 3).
Distributive property
Distributive property
a(b + c) = ab + ac and a(b − c) = ab − ac
a
the factor outside the parentheses, with its sign
b, c
the terms inside; each one gets multiplied by a
Mental math uses it all the time: 7 × 98 = 7(100 − 2) = 700 − 14 = 686.
Distributive Property examples, step by step
Example 1: Mental math
Warm-up
Use the distributive property to find 6 × 27.
Split 27 into friendly parts: 27 = 20 + 7.
Distribute: 6(20 + 7) = 6 × 20 + 6 × 7.
120 + 42 = 162.
Answer:6 × 27 = 162.
Example 2: Distribute a negative
Core
Expand −4(3x − 2).
Multiply −4 by the first term: −4 × 3x = −12x.
Multiply −4 by the second term, −2: −4 × (−2) = 8.
Factoring runs the distributive property backward: 6x + 9 = 3(2x + 3). Distributing goes from 3(2x + 3) to 6x + 9.
Quick check: 3 questions
Question 1 of 3
Expand 4(x + 3).
Show answer and explanation
Answer: B. Multiply 4 by both terms: 4x + 12. Choice A multiplies only the first term, choice C only the second, and choice D adds 4 to 3.
Question 2 of 3
Expand −3(2y − 5).
Show answer and explanation
Answer: D.−3 × 2y = −6y and −3 × (−5) = 15. Choice A misses the double negative, choice B drops the first sign, and choice C forgets to multiply the 5.
Question 3 of 3
Which shows 18 + 24 rewritten with the greatest common factor?
Show answer and explanation
Answer: A. The GCF of 18 and 24 is 6, and 6(3 + 4) = 18 + 24 = 42. Choice B is true but uses a smaller common factor, and choices C and D do not equal 42.
Distributive Property: frequently asked questions
How do you use the distributive property?
Multiply the number outside the parentheses by each term inside, keeping each term’s sign, then add the results. For 5(2x − 3): 5 × 2x = 10x and 5 × (−3) = −15, so the answer is 10x − 15. Then combine any like terms that appear.
Does the distributive property work with subtraction and division?
Yes for subtraction: a(b − c) = ab − ac, as in 4(10 − 1) = 40 − 4 = 36. Division of a sum by a number also distributes: 12 + 84 = 3 + 2. But a number divided by a sum does not: 124 + 2 is 2, not 3 + 6.
Why is it called the distributive property?
To distribute means to hand out to each member of a group. The factor outside the parentheses is handed out to every term inside, the way a teacher hands one worksheet to each student. That is why students often say they ‘distribute the 3’ in 3(x + 4).
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 6.EE.A.3, CCSS 3.OA.B.5, CCSS 6.NS.B.4
Cite this entry
Effortless Math Team. “Distributive Property.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/distributive-property/
Flashcard
TermDistributive Propertya(b + c) = ?
Definitiona(b + c) = ab + ac: multiply the outside factor by each term inside. 3(x + 4) = 3x + 12.
Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2SubtractionSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2, Grade 3FactoringFactoring rewrites a number or expression as a product of factors. It is the reverse of multiplying or distributing: 12 = 3 × 4, 6x + 9 = 3(2x + 3) and x2 + 5x + 6 = (x + 2)(x + 3). Factoring changes the form, not the value, so multiplying the factors back gives the original.Algebra · Grade 6, Grade 7, Grade 8NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2BinomialA binomial is a polynomial with exactly two terms, joined by a plus or minus sign. Examples are x + 7, 3m − 4 and a2 − 9. Combine like terms before counting: 3x + 5x is 8x, one term. A polynomial with one term is a monomial; three terms make a trinomial.Algebra · Algebra 1, Algebra 2SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Commutative PropertyThe commutative property says that changing the order of the numbers does not change the result of adding or multiplying them: a + b = b + a and a × b = b × a. For example, 4 + 9 = 9 + 4 and 3 × 5 = 5 × 3. Subtraction and division are not commutative.Operations · Grade 1, Grade 3, Grade 6Associative PropertyThe associative property says that when you add or multiply three or more numbers, the way you group them does not change the result: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c). It lets you pick the easiest pair first. It does not work for subtraction or division.Operations · Grade 1, Grade 3, Grade 6Like TermsLike terms have exactly the same variables raised to exactly the same powers; only their coefficients may differ. 3x and 8x are like terms, and so are 5a2 and −2a2, but 4x and 4y are not. Only like terms can be combined by adding or subtracting their coefficients: 3x + 8x = 11x.Algebra · Grade 6, Grade 7, Grade 8DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5ExpressionAn expression is a number, variable, or combination of numbers and variables joined by operations, without an equal sign. Examples are 7, 4x + 9 and 3(a − 2). Expressions can be simplified, but they are not solved unless they are part of an equation. You evaluate one by replacing each variable with a number.Algebra · Grade 6, Grade 7, Grade 8EquationAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Algebra · Grade 6, Grade 7, Grade 8Greatest Common FactorThe greatest common factor (GCF) is the largest factor shared by two or more numbers: the largest number that divides into both without a remainder. The GCF of 18 and 24 is 6. Find it by listing factors or by multiplying the prime factors the numbers share. The GCF is how you simplify fractions in one step.Number System · Grade 5, Grade 6, Grade 7