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Reducing Fractions

Reference > Mathematics > Algebra > Algebraic Fractions

After teaching mathematics for many years, I can say with some authority that very few students like fractions. Fractions scare and intimidate students, and the notion of inserting variables into something that already freaks them out can be quite intimidating. In this unit we'll explore algebraic fractions (or, as they're often called, "rational expressions"). My hope is that we can closely relate algebraic fractions to the lessons you learned about fractions in elementary and middle school, so the introduction of variables will be a little less stressful.

With that in mind, for every operation I teach you, we're going to start by focusing on exactly the types of problems you're already familiar with: performing the operation on numerical fractions. I'm going to reteach you how to manipulate numerical fractions, not because you don't know how to do it, but because I want you to start thinking of fractions in a very particular way which will help you later on when we start throwing variables into the mix. Ready?

In studying fractions, there is a word that I want you to drill into your brain repeatedly, until there's no chance of forgetting it. That word is: FACTOR. Any time you see a fraction, you should FACTOR both the numerator and the denominator. That factoring may (at times) seem redundant and unnecessary, but thinking in terms of factoring is going to help you when we get to algebraic fractions. So, let's get started with...

Reducing Fractions

Suppose you want to reduce the fraction

, what would you do? It's fairly obvious that both 6 and 4 are divisible by 2, so you can cancel out a 2, leaving

. What I'd like you to think, though, is "what is the factorization of 6?" and "what is the factorization of 4?"

2·3

2·2

One of our fundamental rules of fractions is that if the numerator is written as a series of factors multiplied together, and the denominator is written as a series of factors multiplied together, and we have a factor that matches in both, then we can divide that factor out of both the numerator and the denominator:

2·3

2·2

2·3

2·2

Now, I get that this seems a bit like overkill, and you're not going to want to do this process, but I still need you to think of fractions in this way. You won't regret it later on. Let's try another fraction.

108

We're going to begin by fully factoring both the 108 and 80: 108 = 2²·3³; 80 = 2⁴·5

108

2²·3³

2⁴·5

In this case, we have a common factor (2), but it has exponents. The numerator only has 2 twos, while the denominator has 4 twos. So how many twos can we cancel? Obviously, we can only cancel 2 of them, since the numerator doesn't have any more than that. And the twos are the only factors that match.

108

2²·3³

2⁴·5

3³

2²·5

IMPORTANT RULE: Every time you see a fraction, you should perform the following two steps: FACTOR and REDUCE*

So how does this work with algebraic fractions? If you are thinking FACTOR and REDUCE, it works exactly the same way.

Example #1

Simplify (reduce)

28x²yz

20xyz³

Solution #1

First, we fully factor the numerator and the denominator: 28x²yz = 2²·7·x²yz; 20xyz³ = 2²·5·xyz³

28x²yz

20xyz³

2²·7x²yz

2²·5·xyz³

What do we see as factors that match in the numerator and denominator? there's a 2² which can cancel. There's also an x, a y, and a z that can cancel. Remember that if a factor matches, but the exponents are different, we can only cancel out the factor raised to the smaller exponent.

28x²yz

20xyz³

2²·7x²yz

2²·5·xyz³

5z²

Now, before we move on to our next problem, there's something important that I need to point out. In dealing with numeric fractions, we never deal with a denominator of zero, because we know that we can't divide by zero, and that means we can't have zero as the denominator (if you don't remember this, try entering 1/0 into your calculator; it'll tell you "Domain Error" or some other similar message)

Now that we've got fractions with variables in them, is it possible to have fractions with zero denominators? Sure! In this example, if either x, y, or z was zero, then that fraction we started with would have had a zero denominator, and therefore, would have been undefined. So, strictly speaking, we should add to our solution that x, y, and z can't be zero. The full answer is:

5z²

, x ≠ 0, y ≠ 0, z ≠ 0.

Example #2

Simplify

x² + 3x + 2

x² - 1

Solution #2

At this point I'd like to make mention of something very important. The fraction bar is a grouping symbol, just like parentheses are a grouping symbol. The fraction bar means "everything above me, taken as a group, divided by everything below me, taken as a group." A lot of students forget this, and they do things like this: "Oh, look! There's an x² up top, and an x² down in the bottom! I'll cancel them!" But remember our rule: FACTOR and REDUCE. To help my students avoid the mistake of canceling before factoring, I tell them they should put extra parentheses in whenever they have a numerator or denominator with more than one term in it. This serves to remind them that the fraction bar is a grouping symbol.

x² + 3x + 2

x² - 1

(x² + 3x + 2)

(x² - 1)

Now it's time to factor. Both the trinomial in the numerator and the binomial in the denominator are factorable:

(x² + 3x + 2)

(x² - 1)

(x + 1)(x + 2)

(x + 1)(x - 1)

Now that we have the numerator and denominator factored, we look for factors that match, and we find one: (x + 1)

(x² + 3x + 2)

(x² - 1)

~~(x + 1)~~(x + 2)

~~(x + 1)~~(x - 1)

(x + 2)

(x - 1)

, x ≠ -1, x ≠ 1 (because those are the values that would make the denominator zero)

Example #3

Simplify

x + 1

x² + 2x + 1

Solution #3

Remember that the numerator and denominator are groups, so let's put those parentheses in to remind ourselves of that fact.

x + 1

x² + 2x + 1

(x + 1)

(x² + 2x + 1)

What do we do next? FACTOR!

(x + 1)

(x² + 2x + 1)

(x + 1)

(x + 1)²

Now we have a matching factor: (x + 1). We can cancel it out of the numerator and denominator. But watch out! There's something very dangerous that students often do; when you cancel the (x + 1) out, there's nothing left in the numerator, which can fool you into thinking it's not a fraction. If you've canceled everything out of the numerator, remember that there's actually a one up there!

(x + 1)

(x² + 2x + 1)

(x + 1)

(x + 1)²

(x + 1)

, x ≠ -1

And that's your lesson on reducing fractions. No problem, right? Just remember FACTOR first! Then REDUCE!

* There is one notable exception to the "reduce" rule, which we'll get to a little bit later on. But for now, drill this into your brain: FACTOR and REDUCE!