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# Distributive Property

Math tricks are often fun ways to transform the numbers you have into the information you need, and the distributive property can be one of the most powerful tricks at your disposal. The distributive property states that for all real numbers x, y, and z the following equation is true:

$x\left(y+z\right)=xy+yz$

That might seem complicated, but it becomes much easier to understand once you sub out the variables for known numbers. For instance:

$3\left(6+7\right)=3\left(13\right)=39$

Here, we added 6 and 7 and multiplied the result by 3 to arrive at the final answer of 39. Alternatively, we can multiply both the 6 and the 7 by 3 and add the resulting products together to reach the same answer:

$3\left(6+7\right)=3\left(6\right)+3\left(7\right)=18+21=39$

Since you get the same answer either way, feel free to choose the methodology that you find easier to work with.

## Why is the distributive property valid in mathematics?

The distributive property is rooted in the distributive law, which says that the equality $x\left(y+z\right)=xy+yz$ is always true in elementary algebra. The proper terminology to describe this relationship is to say that multiplication distributes over addition, which is a fancy way of saying that you can eliminate the parentheses in the example above by multiplying the numbers inside of them by the numbers outside of them.

The distributive law applies to multiple algebraic structures and has applications across multiple levels of mathematics, making the distributive property very important. It is also a building block for more advanced concepts in mathematics.

## Using the distributive property to help with mental math

Performing multiplication in your head is easy enough if the numbers involved are small, but larger numbers might give you a headache. For example, can you multiply 7 x 997 without scrap paper or a calculator? Probably not.

Luckily, the distributive property can make problems like this easier. You can sometimes use the distributive property to express complex multiplication problems as two or more simpler problems that you can do in your head. For example, we can express $7×997=7×\left(1000-3\right)$ , putting it in the same format as the example above.

Multiplying 7 by 1,000 is still a big number, but working with increments of 1,000 is much easier to do in your head than numbers like 997. That means we're working with $7×\left(1000\right)-7×\left(3\right)$ , or $7000-21$ . The final answer is $6979$ , and we got there much more easily than we would have if we were trying to work with an awkward number such as $997$ .

Let's look at another example: $1309×3$ . We can express this as $3\left(1000+300+9\right)$ , giving us $3\left(1000\right)+3\left(300\right)+3\left(9\right)$ , or $3000+900+27$ . The final answer is 3,927, and again we didn't have to worry about working with a number such as 1,309. You cannot simplify every possible multiplication question this way, but it comes up more often than you might expect and can potentially save you a ton of time.

## Using the distributive property to simplify algebraic expressions

Simplifying algebraic expressions is another important use case for the distributive property. Consider the following equation:

$7p+3q-21p+8q$

That's a lot of variables to work with! Fortunately, the distributive property allows us to combine like terms and end up with something that looks like this:

$\left(7-21\right)p+\left(3+8\right)q$

That works out to $-14p+11q$ , an expression that's much simpler than the one we started with. Let's take a look at another example:

$4x+18y-3x-7x+3xy$

The distributive property can help us simplify this equation as well, but it also reveals some of its limitations. The distributive property can only be applied when the variables are exactly the same, so we can combine the $4x$ , $-3x$ , and $-7x$ to get $-6x$ but cannot do anything with the $18y$ or the $3xy$ . That yields the following expression:

$-6x+18y+3xy$

The result is a little more complicated than our first example but still simplified from its original form thanks to the distributive property. Basically, you can add terms with variables together as long as the variables are the same. If there's even a tiny difference (such as the x vs. xy above), you have to leave them alone in your final answer.

It may even be possible to eliminate variables entirely this way. Consider the following expression:

$12xy+9y-8x-9y$

We don't precisely know what y represents, but we do know that we're adding and subtracting 9 times its value in this expression. They are going to cancel each other out no matter what the underlying value is, allowing us to simplify the expression to:

$12xy-8x$

You're ultimately saving yourself time and energy by eliminating variables or just expressing them in simpler terms, so you'll want to apply the distributive property whenever you are able to. Of course, understanding what to do on a math test is a good feeling too!

## Applying the distributive property on binomials through FOIL

If you're multiplying binomials, you can use the FOIL method as a shortcut to applying the distributive property. FOIL is an acronym that stands for First, Outer, Inner, Last and ensures that products are never repeated. Consider the following:

$\left(2x+3\right)\left(x+4\right)$

You'll always get four products when multiplying binomials using the FOIL method. "First" means that we multiply both numbers on the left side of the parentheses, giving us 2x2. "Outer" means multiplying the left side of the first parentheses with the right side of the second, giving us 8x. "Inner" is the right side of the first parentheses and the left side of the second, giving us 3x. Finally, "Last" means that we multiply both numbers on the right side to 12. That gives us:

$2{x}^{2}+8x+3x+12$

While we'll always get four products, we may end up with like terms that can be combined. In the example above, 8x and 3x can be combined for a final answer of:

$2{x}^{2}+11x+12$

Remember that the FOIL method is only applicable when you're working with two binomials. If you're working with other polynomials, you'll have to apply the distributive property without relying on FOIL.

## Other uses of the distributive property

You'll probably stick to applying the distributive property to real numbers and relatively simple expressions right now, but advanced students of mathematics apply its basic principles to many other topics including complex numbers, polynomials, matrices, rings, and fields. Furthermore, college-level math such as Boolean algebra and mathematical logic makes extensive use of the underlying principles of the distributive property even if they don't always distribute basic operations such as addition and multiplication. Furthermore, multiple laws and theorems in the field of mathematics make extensive use of the distributive property as part of broader proofs and applications.

If you're interested in mathematics, understanding the distributive property now could give you a leg up on future courses and help prepare you for an exciting career in a STEM field. If math isn't really your thing, you should still try and understand it because you aren't leaving it behind any time soon. Most students study math through at least high school where concepts such as complex numbers and polynomials are covered.

## Distributive property practice problems

a. $5×4996=?$ Use the distributive property instead of scrap paper.

$5\left(5000\right)-5\left(4\right)=24980$

b. $3×\left(3001\right)$ = ? Use the distributive property instead of scrap paper.

$3\left(3000\right)-3\left(4\right)=9003$

c. Use the distributive property to simplify the following expression:

$3xy+17x-4y-9xy+5x$

$-6xy+22x-4y$

d. Use the distributive property to simplify the following expression:

$27n-13x+9y+13x-8n$

$19n+9y$

e. An $i$ is used to represent the imaginary unit $\sqrt{-1}$ in the study of complex numbers. Using the distributive property, simplify the following expression treating $i$ like any other variable.

$18i+9-11i+30x$

$7i+9+30x$

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