Concept Overview
The distributive property allows multiplication to be applied across terms inside parentheses. It supports algebraic simplification, solving equations, and factoring. The property works in two directions: expanding an expression and factoring to reverse that process.
Key Vocabulary
- Distribute: To multiply a term outside parentheses by each term inside the parentheses.
- Factor: To rewrite an expression as a product of its components.
- Term: A single number, variable, or product of numbers and variables.
- Equivalent Expressions: Expressions that have the same value for all inputs.
Expanding with the Distributive Property
To expand an expression means to apply multiplication to all terms inside parentheses:
\[ a(b + c) = ab + ac \]
Example: Expand the Expression
Expand \( 3(x + 4) \):
\( 3(x + 4) = 3x + 12 \)
Every term inside the parentheses is multiplied by 3.
Prime Factorization
When factoring numerical expressions, it helps to understand how numbers break down into products of prime numbers. This foundation supports identifying greatest common factors.
Example: Prime Factorization
The prime factorization of 18 is \( 2 \times 3 \times 3 \), or \( 2 \times 3^2 \).
The prime factorization of 24 is \( 2 \times 2 \times 2 \times 3 \), or \( 2^3 \times 3 \).
The greatest common factor (GCF) of 18 and 24 is \( 2 \times 3 = 6 \).
Recognizing prime factors helps determine the largest common term that can be factored from expressions like:
\[ 18x + 24 = 6(3x + 4) \]
Factoring Using the Distributive Property
Factoring is the reverse of expanding. Look for a common factor in each term and rewrite the expression as a product:
\[ ab + ac = a(b + c) \]
Example: Factor the Expression
Factor \( 5x + 10 \):
Both terms share a factor of 5: \( 5x + 10 = 5(x + 2) \)
Try It Yourself!
- Expand: \( 6(y – 2) \)
- Factor: \( 12a + 18 \)
- Expand: \( -5(x + 3) \)
Reveal Answers
- \( 6y – 12 \)
- \( 6(2a + 3) \)
- \( -5x – 15 \)
Connections & Extensions
The distributive property appears in solving equations, simplifying polynomial expressions, and factoring quadratics. It is also used when finding equivalent expressions, modeling area, and working with functions.