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Distributive Property

Suppose we have 3 baskets, each holding 2 apples and 4 oranges.

This is the same number of apples and oranges as if we had a bag with 6 apples and a bag with 12 oranges. Except that we now mysteriously no longer have our three baskets, which were handmade in Santa Fe and actually hold quite a bit of sentimental value for us. That's a shame.

Regardless of how we package them, the number of fruits remains the same. Not that it takes away the sting of having had our wicker receptacles stolen from right under our noses.

This is an example of the distributive property, which basically says that it doesn't matter how we "package" numbers when performing multiplication. To write the distributive property in symbols, we say that if ab, and c are real numbers, then:

a(b + c) = ab + ac

When we go from the left side to the right side of this equation, we say we're "distributing a over the quantity (b + c)." We may not say that aloud often, but we certainly won't hesitate to type it. In fact, we just did. 

Sample Problem

Multiply 3x(x + 2).

Now the thing we're distributing is 3x, rather than a plain old number all by its lonesome. That's okay, because the distributive property still works.

3x(x + 2) =
3x(x) + 3x(2)

Remember how exponent notation works? If not, get a refresher here. If we distribute something that has a variable over a quantity in parentheses that also contains that variable, we use exponent notation to keep things tidy. Probably couldn't hurt to also spray it down with a few squirts of Glass Plus. Let's use exponents to finish up.

3x(x) + 3x(2) =
3x2 + 6x

Sample Problem

Expand -2(a + b).

Be careful: When the value you're distributing has a negative sign, make sure you distribute the negative sign over everything in the parentheses. Your parents may have told you to stop spreading your negativity, but ignore them for now.

-2(a + b) = -2a – 2b

Ah, much better.

Sample Problem

Expand -(c + d). 

Having a negative sign by itself outside the parentheses is the same as having -1 outside the parentheses. The 1 is there; it's just hiding. Did you check under the bed? That's totally its favorite spot. To distribute the negative sign, you would simply multiply each term inside the parentheses by -1.

-(c + d) = -cd

Sample Problem

What's the expanded version of -(2a – 5b – 6 + 11c)?

Just multiply every stinkin' term inside those parentheses by -1.

-(2a – 5b – 6 + 11c) = -2a + 5b + 6 – 11c

By the way, since multiplication is commutative, the distributive property also works if we write the multiplication the other way around:

(b + c)a = ba + ca.

Sample Problem

Use the distributive property to multiply (4xy)(-3).

This is the same thing as -3(4xy), so just multiply -3 by both terms and voila:

(4xy)(-3) =
4x(-3) – y(-3) =
-12x + 3y

Sample Problem

Use the distributive property to multiply (4 – x)(-1).

Same old, same old. Tack a -1 onto both terms.

(4 – x)(-1) =
4(-1) – x(-1) =
-4 + x

Ready to really get down to business? The distributive property still works even if the expression in parentheses has more than two terms. It's totally a team player.

Sample Problem

What's the expanded version of 4(x + y + z)?

This one's not too bad. Slap a 4 onto each variable and we're done.

4(x + y + z) = 4x + 4y + 4z

The distributive property also works when we're multiplying expressions where both factors have multiple terms. So if you're a tennis player, it's like playing straight doubles rather than Canadian doubles. Or triples. Okay, the analogy sort of falls apart at this point. Ignore us and take a look at one more example.

Sample Problem

Expand (3 + x)(y – 4).

Okay, for this one we'll use the distributive property twice. It's double-distributing time. Basically, we want to keep on distributin' until the day is done. Or at least until there's nothing left to distribute.

First we separate the 3 and the x in the first factor.

(3 + x)(y – 4) =
3(y – 4) + x(y – 4)

Then we distribute both terms separately like normal.

3(y – 4) + x(y – 4) =
3y – 12 + xy – 4x

Man, that's a lot of stuff to keep track of at once. Think how much easier it would be if you could shower, brush your teeth, eat breakfast, and get dressed all at the same time. What a life-saver that would be! Especially on mornings that your alarm didn't go off...

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