Distributive Property
The distributive property says that if you multiply a term times multiple other terms (this usually looks like a term multiplied by a polynomial in parentheses) you multiply the first term by each term in the parentheses, and then combine the products. Essentially, you get rid of the parentheses by "distributing" the multiplier term to each term inside the parentheses.
According to the order of operations, when you have parentheses, you combine the terms in the parentheses first, before you do any other operations. But, if the terms inside the parentheses are not alike, you can't combine them. That's when distributive property comes in. If a term is multiplied times a parenthetical term (remember, in algebra, when a term is right up against another term, with no operating symbol, it means multiplication), you "distribute" the term times every single term inside the parentheses. Then you combine like terms to simplify.
Example:
$$\eqalign{3(4x+2)&=\\\overrightarrow{\overrightarrow{3(}4x +}2)&=\qquad && \text{Distribute the three to all terms in the parentheses}\\(3\times4x)+ (3 \times2)&=\qquad&& \text{Distribute means multiply by each term, join multiplication problems with same operation}\\12x+6&=12x+6\qquad&&\text{Simplify}}$$
We most commonly think of distributive property problems that look like the one above but the multiplier terms can have variables and there can be many terms in the parentheses. The same rules apply: multiply the multiplier (the term you are multiplying times that parenthetical term) by each term inside the parentheses:
Example:
$$\eqalign{5x(3x^2+2x-6y+8)&=\\\overrightarrow{\overrightarrow{\overrightarrow{\overrightarrow{5x(}3x^2 +}2x-}6y+}8)&=\qquad && \text{Distribute the 5x to all terms in the parentheses}\\(5x\times3x^2)+ (5x \times2x)+(5x\times -6y)+(5x\times8)&=\qquad&& \text{Distribute means multiply by each term}\\15x^3+10x^2-30xy+40x&=\qquad&&\text{Simplify}}$$
Overall, when you see a term multiplied by a parenthetical, just multiply the term by each term in the parenthetical and combine.
Practice Problems:
Distributive Property
Use the distributive property to evaluate or simplify the expression.
1. $-6(-4+9)$
2. $5(b-3)$
3. $-2(-5-4)$
4. $-12(b-8)$
5. $4(2m-3)$
6. $-3(6-2r)$
7. $-1(3w-4)$
8. $2(-p+4)$
9. $-5(11-3)$
10. $-2(-2n+1)$
11. $4(z-3)=4$
12. $-7(x+2)=7$
13. $\dfrac{1}{2}(w-3)=\dfrac{-3}{2}$
14. $5(3w-2)=5$
15. $10(3x+2)=5$
16. $.5(1-4y)=2$
17. $7(5 +(-8x))=91$
18. $\dfrac{3}{4}(2x+\dfrac{2}{3})=4$
19. $15(2w+\dfrac{1}{3}w)=-70$
20. $\dfrac{3}{2}(\dfrac{1}{2}r-6)=\dfrac{r}{4}-5$
Answer Key:
Test Prep Practice
Distributive Property Test Prep
1. If $5(y-4)=3(y-4)$, what is the value of $y$?
(A)4
(B)16
(C)-16
(D)-4
(E)0
2. If $10x-10y=100$, what is the value of $5(10x-10y)$?
(A)100
(B)-100
(C)500
(D)-500
(E)5
3. If $a(x-y)=81$ and $ax=36$, what is the value of $ay$?
(A)117
(B)54
(C) 45
(D)-45
(E)-117
4. If $(n-3)(8-15)=-60$, then $n=$
Grid in your answers
5. $3x(4x+2)+6(4x+2)=ax^2+bx+c$
In the equation above, $a$, $b$, and $c$ are constants. If the equation is true for all values of x, what is the value of b?
Grid in your answers
Answer Key: