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Before you get started, take this readiness quiz.
Suppose three friends are going to the movies. They each need $\text{\$9.25};$ that is, $9$ dollars and $1$ quarter. How much money do they need all together? You can think about the dollars separately from the quarters.
They need $3$ times $\text{\$9},$ so $\text{\$27},$ and $3$ times $1$ quarter, so $75$ cents. In total, they need $\text{\$27.75}.$
If you think about doing the math in this way, you are using the Distributive Property.
If $a,b,c$ are real numbers, then
Back to our friends at the movies, we could show the math steps we take to find the total amount of money they need like this:
In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression $3(x+4),$ the order of operations says to work in the parentheses first. But we cannot add $x$ and $4,$ since they are not like terms. So we use the Distributive Property, as shown in [link] .
Simplify: $3(x+4).$
$3(x+4)$ | |
Distribute. | $3\xb7x+3\xb74$ |
Multiply. | $3x+12$ |
Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in [link] would look like this:
The distributive property can be used to simplify expressions that look slightly different from $a(b+c).$ Here are two other forms.
If $a,b,c$ are real numbers, then | $a(b+c)=ab+ac$ |
Other forms: |
$a(b-c)=ab-ac$
$(b+c)a=ba+ca$ |
Do you remember how to multiply a fraction by a whole number? We’ll need to do that in the next two examples.
Simplify: $\frac{3}{4}(n+12).$
Distribute. | |
Simplify. |
Simplify: $8\left(\frac{3}{8}x+\frac{1}{4}\right).$
Distribute. | |
Multiply. |
Simplify: $12\left(\frac{1}{3}n+\frac{3}{4}\right).$
4 n + 9
Using the Distributive Property as shown in the next example will be very useful when we solve money applications later.
In the next example we’ll multiply by a variable. We’ll need to do this in a later chapter.
Simplify: $m(n-4).$
Distribute. | |
Multiply. |
Notice that we wrote $m\xb74\phantom{\rule{0.2em}{0ex}}\text{as}\phantom{\rule{0.2em}{0ex}}4m.$ We can do this because of the Commutative Property of Multiplication. When a term is the product of a number and a variable, we write the number first.
The next example will use the ‘backwards’ form of the Distributive Property, $(b+c)a=ba+ca.$
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