Distributive Property - Basic Forms
The Distributive Property allows multiplying a number by a sum or difference by multiplying each term inside the parentheses separately. From OpenStax Elementary Algebra 2E, Chapter 1, the forms are: a(b+c)=ab+ac and a(b-c)=ab-ac. For 7(x+2)=7x+14, and for -5(y-4)=(-5)(y)+(-5)(-4)=-5y+20. The Distributive Property is used to expand expressions, simplify equations, and is the foundation for polynomial multiplication. It applies with positive, negative, fractional, and variable coefficients.
Key Concepts
Property If $a, b, c$ are real numbers, then $a(b+c) = ab+ac$ $(b+c)a = ba+ca$ $a(b c) = ab ac$ $(b c)a = ba ca$.
Examples To simplify $7(x+2)$, distribute the $7$: $7 \cdot x + 7 \cdot 2 = 7x + 14$. To simplify $ 5(y 4)$, distribute the $ 5$: $( 5) \cdot y ( 5) \cdot 4 = 5y + 20$. To simplify $9 3(x+1)$, first distribute the $ 3$: $9 3x 3$. Then combine like terms to get $6 3x$.
Explanation The distributive property lets you 'distribute' or 'pass out' the number outside the parentheses to every term inside. It's the key to removing parentheses and simplifying expressions in algebra. Think of it as sharing the multiplication.
Common Questions
What does the Distributive Property state?
a(b+c) = ab+ac and a(b-c) = ab-ac. You multiply the factor outside the parentheses by each term inside separately.
Distribute 7(x + 2).
7(x)+7(2) = 7x+14.
Distribute -5(y - 4).
-5(y)+(-5)(-4) = -5y+20.
How does the Distributive Property apply when distributing a negative sign?
Distributing a negative sign means multiplying each term inside by -1: -(a+b) = -a-b. Every sign inside the parentheses changes.
Use the Distributive Property to simplify 3(2x - 5) + 4.
Distribute: 6x-15+4. Combine like terms: 6x-11.