The Distributive Property: Definition, Examples and Practice Problems
Part of the Math Concepts guide for Grades 3 to 8.
What Is the Distributive Property?
The distributive property states that multiplying a number by a group of numbers added together gives the same result as multiplying each number separately and then adding. In algebra it is written as: a(b + c) = ab + ac. The key idea is that the multiplication is distributed across each term inside the parentheses.
Key Formulas
| Form | Formula | Example |
|---|---|---|
| Multiplication over addition | a(b + c) = ab + ac | 3(4 + 5) = 12 + 15 = 27 |
| Multiplication over subtraction | a(b − c) = ab − ac | 6(10 − 3) = 60 − 18 = 42 |
| Reverse (factoring) | ab + ac = a(b + c) | 6x + 15 = 3(2x + 5) |
Step-by-Step Examples
Example 1: Basic (Grade 3)
Problem: 5 x (3 + 4)
- Identify the number outside the parentheses: 5
- Multiply 5 by each term inside: 5 x 3 = 15, then 5 x 4 = 20
- Add the products: 15 + 20 = 35
- Check: 5 x 7 = 35. Correct!
Answer: 35
Example 2: Subtraction variant (Grade 4)
Problem: 7 x (9 − 4)
- Multiply 7 by 9: 63
- Multiply 7 by 4: 28
- Subtract: 63 − 28 = 35
- Check: 7 x 5 = 35. Correct!
Answer: 35
Example 3: With variables (Grade 6)
Problem: Expand 4(x + 6)
- Multiply 4 by x: 4x
- Multiply 4 by 6: 24
- Write the sum: 4x + 24
Answer: 4(x + 6) = 4x + 24
Example 4: Negative sign (Grade 7)
Problem: Expand 3(2x − 5)
- Multiply 3 by 2x: 6x
- Multiply 3 by −5: −15. Keep the negative sign!
- Write the result: 6x − 15
Answer: 3(2x − 5) = 6x − 15
Example 5: Factoring (reverse) (Grade 7)
Problem: Factor 12y + 8
- Find the GCF of 12 and 8: GCF = 4
- Factor 4 from each term: 4 x 3y = 12y, and 4 x 2 = 8
- Write in factored form: 4(3y + 2)
Answer: 12y + 8 = 4(3y + 2). You can check this by distributing: 4(3y + 2) = 12y + 8.
Using the Distributive Property to Remove Parentheses
One of the most common uses of this property in middle school algebra is expanding expressions. Here is a clear step-by-step process students can follow every time.
- Identify the number or term directly outside the parentheses.
- Multiply that term by the first term inside the parentheses.
- Multiply that term by the second term inside the parentheses. Watch out for signs!
- Write the two products connected by the same operation (+ or −) that was inside the parentheses.
For example, to expand 5(2x + 3): multiply 5 by 2x to get 10x, then multiply 5 by 3 to get 15. Result: 10x + 15. No parentheses needed.
Common Mistakes to Avoid
Mistake 1: Only distributing to the first term
2(x + 5) does not equal 2x + 5. You must multiply BOTH terms inside the parentheses: 2(x + 5) = 2x + 10. This is the most common error I see from students first learning this property.
Mistake 2: Forgetting to flip signs with a negative multiplier
−3(x − 4) = −3x + 12. Distributing a negative number flips all the signs inside. The − outside changes both + and − signs inside. Students often get the first term right but forget to flip the sign on the second term.
Mistake 3: Applying it to exponents
The distributive property only works for multiplication over addition and subtraction. It does not work for exponents. (x + y)² does not equal x² + y². That is a very common mistake in algebra that leads to errors all the way through quadratics.
Common Questions About the Distributive Property
What is the distributive property in simple terms?
The distributive property means you can multiply a number by everything inside a set of parentheses, one term at a time, and get the same answer as if you had added everything first. For example, 4 x (3 + 2) = (4 x 3) + (4 x 2) = 20, which is the same as 4 x 5 = 20.
How do you use the distributive property to remove parentheses?
Multiply the number outside the parentheses by every term inside, one at a time. For 5(2x + 3): multiply 5 x 2x = 10x, then 5 x 3 = 15. The result is 10x + 15 with no parentheses.
What is the distributive property of multiplication?
The distributive property of multiplication states that a x (b + c) = (a x b) + (a x c). This works for any real numbers and is one of the fundamental properties of arithmetic and algebra.
What is the distributive property of addition?
In reverse (factoring), the distributive property means ab + ac = a(b + c). For example, 6x + 10 = 2(3x + 5). This is the same property read from right to left.
Does the distributive property work with subtraction?
Yes. The full form is a(b − c) = ab − ac. For example, 8(x − 3) = 8x − 24. The key is to keep the minus sign: when you multiply by a negative term, the result is negative.
What grade do students learn the distributive property?
Students are first introduced to the distributive property in 3rd grade through the idea of breaking apart multiplication. By 6th grade it is applied formally to algebraic expressions. It continues to appear throughout middle and high school algebra.
