Lesson 3: Distributive Property and Common Factoring
Lesson 3: Distributive Property and Common Factoring
Expand with the distributive property, then reverse it to factor.
In this lesson we are discovering the distributive property of multiplication, as well as common factoring.
According to the distributive property, a(b + c) = ab + ac.
When multiplying the sum or difference of two terms by another term, it is possible to find the product of each pair of terms first and then add/subtract them.
This is how we expand brackets and simplify algebraic expressions.
Distributive property is used to simplify expressions, solve equations and word problems. It is a part of virtually every math course in high school and the skill is transferable to other concepts.
The opposite (inverse) operation to distribution is common factoring.
This is when we are given the expanded form of an algebraic expression and need to determine the Greatest Common Factor, that all the terms of the expression are divisible by – the term that goes outside of the brackets.
For a more detailed explanation of these processes with examples watch the video lesson above, then check out the accompanying note and complete additional math practice on this topic.
Distributive Property Matching Activity
Key ideas: Distributive Property and Common Factoring
The distributive property says that multiplying a sum (or difference) by a term is the same as multiplying each part separately: a(b + c) = ab + ac and a(b − c) = ab − ac. We use it to expand brackets.
Common factoring is the reverse. Find the greatest common factor (GCF) of all the terms, write it outside the brackets, and divide each term by it to find what goes inside. For the variable part of the GCF, take each variable that appears in every term, with its lowest exponent.
You can always check factoring by expanding your answer: you should get back the original expression.
Worked examples
Example 1
Expand 4(3x − 5).
- 4 × 3x = 12x and 4 × (−5) = −20.
- Answer: 12x − 20
Example 2
Expand and simplify 2x(x + 3) − 5(x − 1).
- 2x(x + 3) = 2x2 + 6x.
- −5(x − 1) = −5x + 5 (a negative times a negative is positive).
- 2x2 + 6x − 5x + 5.
- Answer: 2x2 + x + 5
Example 3
Factor 6x2 + 9x.
- GCF of 6 and 9 is 3; both terms contain x, lowest power x1. GCF = 3x.
- 6x2 ÷ 3x = 2x and 9x ÷ 3x = 3.
- Answer: 3x(2x + 3). Check: 3x(2x + 3) = 6x2 + 9x ✓
Example 4
Factor 12a3b2 − 18a2b + 6ab.
- GCF of 12, 18, 6 is 6. Lowest powers: a1 and b1. GCF = 6ab.
- Divide each term: 2a2b, −3a and +1.
- Answer: 6ab(2a2b − 3a + 1)
Common mistakes
- Multiplying only the first term: 3(x + 4) is 3x + 12, not 3x + 4.
- Sign errors with a negative factor: −2(x − 4) = −2x + 8.
- Not taking out the greatest factor: 3(2x2 + 3x) is not fully factored; 3x(2x + 3) is.
- Dropping the 1 when a term equals the GCF: 6ab(2a2b − 3a + 1), not 6ab(2a2b − 3a).
Questions students ask
How do I know my factoring is complete?
Check that the terms left inside the brackets have no common factor other than 1.
Why is factoring useful?
It simplifies expressions and fractions, and later it is the key step for solving quadratic equations.
Check your understanding
Try the quiz for this lesson. You’ll see right away if each answer is correct.