The distributive property is a rule that connects multiplication and addition. It says that multiplying a number by a sum gives the same result as multiplying that number by each addend and then adding the products. This property matters because it helps students simplify expressions, do mental math, and understand why algebraic steps work.
It is one of the main tools for moving between expanded and factored forms.
Key Facts
- Distributive property over addition: a(b + c) = ab + ac
- Distributive property over subtraction: a(b - c) = ab - ac
- Expanded form separates one product into two products, such as 4(x + 3) = 4x + 12
- Factored form reverses distribution, such as 6x + 18 = 6(x + 3)
- An array or area model shows a rectangle split into parts, so total area = ab + ac
- The distributive property works with numbers, variables, fractions, decimals, and negative values
Vocabulary
- Distributive property
- A rule that lets a factor multiply each term inside parentheses, such as a(b + c) = ab + ac.
- Factor
- A number or expression that is multiplied by another number or expression.
- Term
- A single number, variable, or product in an expression, separated by plus or minus signs.
- Expanded form
- An expression written after multiplication has been distributed across terms.
- Factored form
- An expression written as a product of a common factor and a sum or difference.
Common Mistakes to Avoid
- Multiplying only the first term inside parentheses is wrong because the outside factor must multiply every term, so 5(x + 2) becomes 5x + 10, not 5x + 2.
- Forgetting the sign of a term is wrong because subtraction or a negative term must stay with that term, so 3(x - 4) becomes 3x - 12.
- Adding inside parentheses before checking the expression can be wrong in algebra because unlike terms cannot always be combined, so 2(x + 5) cannot become 2(5x).
- Factoring out a number that is not common to all terms is wrong because every term must be divisible by the factor, so 8x + 10 cannot be factored as 4(2x + 3).
Practice Questions
- 1 Expand and simplify: 7(4 + 9).
- 2 Expand the expression: 5(x + 6).
- 3 Use the distributive property to explain why 12(20 + 3) is easier to compute mentally than 12 x 23.