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Distributive Properties

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, New Mexico 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 that will take 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, for all real numbers ab, and c,
and a(b + c) = ab + ac.

When we go from the left side to the right side of this equation, we say we are "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 Problems

1. Multiply: 3x(x + 2).

Now the thing we are 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 + 6x = 3x2 + 6x.

2. - 2(a + b) = - 2a - 2b.

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

Having a negative sign by itself outside the parentheses is the same as having - 1 outside the parentheses. The 1 is there; it is 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. 

Sample Problems

1.  - (c + d) = - c - d.

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

Since multiplication is commutative, the distributive property also works if we write the multiplication the other way around. For all real numbers a, b, and c,

(b + c)a = ba + ca.

Sample Problems

1. Use the distributive property to multiply (4x - y)( - 3).

(4x - y)( - 3) = - 12x + 3y.

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

(4 - x)( - 1) = - 4 + x.

Notice that, in the previous example, we needed to write out the negative one, since the expression (4 - x) -  doesn't make sense. No hiding under the bed for him today

The distributive property works even if the expression in parentheses has more than two terms. It is totally a team player.

Sample Problems

1. 4(x + y + z) = 4x + 4y + 4z.

The distributive property also works to multiply expressions where both factors have multiple terms. So, if you are a tennis player, it is 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 these examples.

2. (3 + x)(y - 4) = (3 + x)y - (3 + x)4, using the distributive property to distribute (3 + x) over (y - 4).

3. (3 + x)(y - 4) = 3(y - 4) + x(y - 4), using the distributive property to distribute (y - 4) over (3 + x).

Shortcut alert! If you are especially on the ball, you may have noticed that we can distribute twice in the cases above. Consider the first example of distributivity. First we can distribute (3 + x) over (y - 4), but afterwards we can actually continue and distribute the y over (3 + x) and distribute the (- 4) over (3 + x). Basically, you want to keep on distributin' until the day is done. Or at least until there is nothing left to distribute.

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

That was a lot of steps! What are we, making our way into the Philadelphia Museum of Art? (Rocky reference, sorry.) Fortunately, there is a shortcut. (No, not for you, Rocky. Keep runnin'.) You may have already heard of it. It is called FOIL. FOIL stands for First, Outer, Inner, Last. The idea here is to be able to carry out both steps of the distribution simultaneously. Aside from helping to keep your baked chicken crispy, FOIL helps you do this stuff in a systematic way so that you don't make mistakes.

1. The First means that you multiply the first term in the first set of parentheses by the first term in the second set of parentheses. Below we put brackets around the first terms. In addition to helping you easily identify the terms in question, putting them in a box also helps guarantee freshness.

([ 3 ] + x)([ y ] + (- 4))

2. The Outer means that you multiply the first time in the first set of parentheses by the second term in the second set of parentheses.

([ 3 ] + x)(y + [ - 4 ])

3. The Inner refers to multiplying the innermost terms:

(3 + [ x ])([ y ] + (- 4))

4. The Last refers to the last terms in each set of parentheses being multiplied:

(3 + [ x ])(y + [ - 4 ])

To carry out the multiplication using FOIL, we multiply the boxed terms in the order presented above:

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

Notice that we arrived at the same answer as we did when we distributed twice, but we did all of the steps at once! Think how much easier it would be if you could shower, brush your teeth, eat breakfast, and get dressed all at once. What a life-saver that would be! Especially on mornings that your alarm didn't go off...

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