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I was learning (practicing to solve) simplifying the rational expressions. I know how to simplify the rational expressions... but I can't understand some part of the questions.

The question that I can't understand

If you look at this image, you could see sentence "First, let's set the denominator equal to zero and solve for n:", and there is "Select all that apply." too.

But, there was no any explanations about it must be denominator = 0. I found it just in hint, not a basic question..

So... why solving equation and suppose that denominator = 0 are related to simplifying the rational expressions?

And why we suppose denominator = 0 and we don't touch numerator..?

Is there any mathematical relationships between them?

I hope you (expert) will answer to my question, thanks.

BrianO
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user298530
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    Are you familiar with the fact that division by zero is undefined for real numbers? You can answer questions like "What is $\frac{5}{2}$?" and "What is $\frac{1}{8}$?", but something like $\frac{6}{0}$ doesn't make sense. Even $\frac{0}{0}$ doesn't make sense. It is for that reason that we figure out when the denominator is zero since we want to make sure not to forget that a rational expression didn't make sense at those points. It is possible that after simplifications, the "not making sense" aspect is forgotten. We don't check the numerator for being zero since $\frac{0}{4}$ is fine. – JMoravitz Dec 14 '15 at 04:36
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    Did you forget that "Which values of $n$ make the expression undefined" was part of the question? – Akiva Weinberger Dec 14 '15 at 05:14
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    Akiva Weinbergerm, Maybe I forgot that... thank you. – user298530 Dec 14 '15 at 05:36

1 Answers1

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The problem is that division by $0$ is undefined. Thus, before we begin simplifying an expression, we want to know when it is undefined. Remember that $0$ divided by something else is just $0$, so we don't need to worry about the numerator.

For example, the expression $\frac{1+x}{x}$ is undefined at $x=0$, but when $x=-1$, we just get $\frac{0}{-1}=0$.

As another example, consider $\frac{x}{x}$. Clearly this is just equal to $1$. However, the fraction is still undefined at $x=0$. That is because $\frac{0}{0}$ is undefined. To compensate, we can write that $\frac{x}{x}$ is equal to $1$ when $x\neq 0$.

user21820
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pancini
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