Without even writing down any algebra, you probably already know what types of problems this article is talking about. There are problems that come up in algebra where it seems like you should be able to cancel a variable out and move on. Yet, in these very situations your professor or teacher keeps telling you it can’t happen. Why is this the case?
Below, we will look at some specific examples and see the difference between when you can cancel a variable and when you can’t.
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It is all about factors
It may not always seem like it, but there really is a pattern and a simple rule of when you can and can’t cancel a term or a variable. Let’s focus on rational expressions specifically. Those are expressions (like the one below) where both the numerator and the denominator are polynomials. An example is:
\(\dfrac{x+5}{x^2+1}\)
When can you cancel terms in a rational expression?
You can cancel any matching factors that occur in both the numerator and the denominator. Let’s use numbers to understand this a little better. Consider the fraction below:
\(\dfrac{4}{14}\)
In the fraction above, 2 is a factor of 4 since \(2 \times 2 = 4\). Similarly, 2 is a factor of 14 since \(2 \times 7 = 14\). Based on the rule of cancelling common factors, we should be able to cancel the 2’s and get a fraction that has the same value.
\(\dfrac{4}{14}=\dfrac{2(2)}{2(7)}=\dfrac{2}{7}\)
If you check on a calculator you will find that 4 divided by 14 is the same as 2 divided by 7 – so this rule seems to be working here. What is really going on?
Why does this work?
Every number other than zero, when divided by itself is 1. Since a fraction simply represents division (in one sense), \(\dfrac{2}{2} = 1\) and we really just have \(1\times\dfrac{2}{7}\).
Trying it with Variables
Now that everything makes some sense with numbers, let’s try variables.You can think of it as though the variables represent numbers, so they will behave in the same way. Take for instance the expression below.
\(\dfrac{x^2-x}{x^3}, x\neq 0\)
The numerator has a couple of factors which we can find by factoring out an \(x\):
\(x^2-x = x(x-1)\)
This shows us that \(x\) and \(x-1\) are factors of the numerator. Looking at the denominator, we see \(x\) is also a factor since \(x^3=x(x^2)\). Remember, when both the numerator and the denominator share a factor, you can cancel that factor from the fraction. This means that we can simplify the original rational expression:
\(\begin{align} \dfrac{x^2-x}{x^3} &= \\ \dfrac{x(x^2-1)}{x(x^2)} &= \boxed{\dfrac{x^2-1}{x^2}}\end{align}\)
The only time you can’t cancel terms in the numerator and denominator is when they are both NOT factors.. That’s why you can’t cancel \(x\) in \(\dfrac{x-5}{x}\). The \(x\) is not a factor of the numerator; its just a term being added. Cancelling the \(x\) here would be like cancelling the \(5\) in \(\dfrac{5+1}{5}\) and saying that is 1, when really \(\dfrac{5+1}{5}\) = \dfrac{6}{5}. This is certainly not equivalent to one!
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Summary
Simplifying rational expressions, like those seen above, requires the same skills you used when simplifying fractions like \(\dfrac{2}{4}\) or \(\dfrac{8}{10}\). You are always looking for common factors. If \(x\) or whatever expression you are looking at isn’t a common factor of the numerator and denominator, then you can’t cancel it out of the rational expression.