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Algebraic Expressions

Algebraic Expressions

At a Glance - Factoring (Distributive Property in Reverse)

We can work the distributive property in reverse—we just need to check our rear view mirror first for small children.

When we rewrite ab + ac as a(b + c), what we're actually doing is factoring. A factor in this case is one of two or more expressions multiplied together. Factoring an expression means breaking the expression down into bits we can multiply together to find the original expression. Why would we want to break something down and then multiply it back together to get what we started with in the first place? Oh, who knows. Those crazy mathematicians have a lot of time on their hands. It actually will come in handy, trust us.

Factoring expressions is pretty similar to factoring numbers. Click here for a refresher. 

Sample Problem

Factor the expression 3x2 – 27xy. Check to see that your answer is correct

Since each term of the expression has a 3x in it (okay, true, the number 27 doesn't have a 3 in it, but the value 27 does), we can factor out 3x:

3x2 – 27xy =
3x(x – 9y)

We can check that our answer is correct by using the distributive property to multiply out 3x(x – 9y), making sure we get the original expression 3x2 – 27xy. We do, and all of the Whos down in Whoville rejoice.

The value 3x in the example above is called a common factor, since it's a factor that both terms have in common. If these two ever find themselves at an uncomfortable office function, at least they'll have something to talk about.

When we factor an expression, we want to pull out the greatest common factor. The greatest common factor is a factor that leaves us with no more factoring left to do; it's the finishing move. That would be great, because as much as we love factoring and would like nothing more than to keep on factoring from now until the dawn of the new year, it's almost our bedtime. Let's find ourselves a GCF and call this one a night. 

Sample Problem

Factor the expression 45x – 9y + 99z.

Right off the bat, we can tell that 3 is a common factor. So let's pull a 3 out of each term.

45x – 9y + 99z =
3(15x – 3y + 33z)

Wait a sec. We can factor this expression even further because all of the terms in parentheses still have a common factor, and 3 isn't the greatest common factor. Although it's still great, in its own way. Really, really great. As great as you can be without being the greatest. This is us desperately trying to save face.

Instead, let's be greedy and pull out a 9 from the original expression.

45x – 9y + 99z =
9(5xy + 11z)

The terms in parentheses have nothing else in common to factor out, and 9 was the greatest common factor. Not that that makes 9 superior or better than 3 in any way; it's just that...well, 3 is simply...oy. Insert foot into mouth.

To find the greatest common factor for an expression, look carefully at all of its terms. The number part of the greatest common factor will be the largest number that divides the number parts of all the terms. The variable part of a greatest common factor can be figured out one variable at a time. Much easier. Both to do and to explain.

For each variable, find the term with the fewest copies. Use that number of copies (powers) of the variable. Finally, multiply together the number part and each variable part. This step will get us to the greatest common factor. 

Sometimes we have a choice of factorizations, depending on where we put the negative signs. We'll show you what we mean; grab a bunch of negative signs and follow us...

Sample Problem

Factor the expression -50x + 4y in two different ways.

First way: factor out 2 from both terms.

-50x + 4y =
2(-25x + 2y)

Second way: factor out -2 from both terms instead.

-50x + 4y =
-2(25x – 2y)

You can double-check both of 'em with the distributive property.

These factorizations are both correct. Neither one is more correct, so let's not get all in a tizzy. Which one you use is merely a matter of personal preference. Or maybe a matter of your teacher's preference, if your teacher asks you to do these problems a certain way. If they do, don't fight them on it. They're bigger than you. Or at least they were a few years ago.

Be Careful: Always check your answers to factorization problems. When you multiply factors together, you should find the original expression. This step is especially important when negative signs are involved, because they can be a tad tricky. In fact, you probably shouldn't trust them with your social security number. Especially if your social has any negatives in it.

Example 1

Factor the expression 2x + 4y. Check to see that your answer is correct.

Example 2

What is the greatest common factor in the expression 4xy – 6x2?

Example 3

What is the greatest common factor in the expression 6xy3 – 9x2y3 + 15y2z?

Example 4

What is the greatest common factor in the expression 3xxy?

Example 5

Factor the expression -8x2z – 16x3. Check your answer.

Exercise 1

What's the greatest common factor of 4xy – 8y + 3y5?

Exercise 2

What's the greatest common factor of 5z + 25z2?

Exercise 3

What's the greatest common factor of abc + 5bc b2c?

Exercise 4

What's the greatest common factor of 20x – 16y?

Exercise 5

Factor 2x3 + 4x as much as possible. Check your answer by multiplication before clicking to see if you got it right. We can't force you to do that, of course, but it would mean a lot to us, and we'd be disappointed in you if you didn't. We'll use the guilt tactic here and see if that works.

Exercise 6

What are the factors of 9y + 6x?

Exercise 7

What are the factors of 10a – 20b + 30c?

Exercise 8

What are the factors of -4z – 12y?

Exercise 9

What are the factors of x2y + xy2?

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