Rational expressions

[Steph Annie]

Hello!

I was taking a math quiz yesterday and came across this problem:
It said, "For what value or values is the rational expression undefined?"
Then x +2/x+5.

I am familiar with rational expressions. You have to make x equal to zero to show that anything divided by zero is undefined.
And then you can choose which side will let you do this. Usually it is the denominator. Right?

But I am used to seeing something like x² + 4 / x² - 3x

So I immediately know to take the denominator expressions of x² - 3x
and say x(x+3)
Then x-3=0
X-3
And x=0

X is undefined for 0,3.

BUT, when I saw the question on the test for x-2/x-5
it wasn't as clear to me which values would make it undefined because they both look the same.
And I only got one value from either.

Like x-5=0
X=5

So is x only undefined for one value?: 5

And if so why not say x-2=0
X=2

I just wasn't sure.

On my quiz, I answered "X is undefined for 0,5."

But I regret putting the zero because that was definitely wrong.

Thanks for any help!

 

[DrPhil]

Hello!

I was taking a math quiz yesterday and came across this problem:
It said, "For what value or values is the rational expression undefined?"
Then x +2/x+5.

I am familiar with rational expressions. You have to make x equal to zero to show that anything divided by zero is undefined.
And then you can choose which side will let you do this. Usually it is the denominator. Right?
ALWAYS the denominator! A zero in the numerator is well defined, but division by zero is not defined.

But I am used to seeing something like x² + 4 / x² - 3x

So I immediately know to take the denominator expressions of x² - 3x
and say x(x+3)
Then x-3=0
X-3
And x=0

X is undefined for 0,3.

BUT, when I saw the question on the test for x-2/x-5
it wasn't as clear to me which values would make it undefined because they both look the same.
And I only got one value from either.

Like x-5=0
X=5

So is x only undefined for one value?: 5 YES. A linear factor has a single zero; the case above is a quadratic with 2 roots.

And if so why not say x-2=0 The numerator MAY be zero.
X=2

I just wasn't sure.

On my quiz, I answered "X is undefined for 0,5."

But I regret putting the zero because that was definitely wrong.

Thanks for any help!

 

[lookagain]

Then (x + 2)/(x + 5).


But I am used to seeing something like (x² + 4)/(x² - 3x)


BUT, when I saw the question on the test for (x - 2)/(x - 5)

Steph Annie, you must have grouping symbols around these numerators and denominators,
such as the use of the parentheses suggested in the quote box above. In future posts,
please include them.

 

[Steph Annie]

Dear LookAgain.

Yes, I realize the importance of showing the parentheses.
However, there were NO parentheses in the original problem on my test, and I've never
see a rational expression with parentheses around the numerator or denominator.
For example, I am looking at the problem from my book: x²+4/X²-3x
and there are NO parentheses.

I will ask my teacher today in class why this is.

Steph Annie, you must have grouping symbols around these numerators and denominators,
such as the use of the parentheses suggested in the quote box above. In future posts,
please include them.

 

[lookagain]

.
For example, I am looking at the problem from my book: x²+4/X²-3x \(\displaystyle \ \ \ \ \) <----- That is wrong in two ways.

1) The grouping symbols are missing.

2) You didn't keep the letter for the variable in the same case.

Regardless, please make the changes I mentioned in your later posts, or the expressions will be wrong,
and they will be equivalent to something different than you intended.

 

[JeffM]

Dear LookAgain.

Yes, I realize the importance of showing the parentheses.
However, there were NO parentheses in the original problem on my test, and I've never
see a rational expression with parentheses around the numerator or denominator.
For example, I am looking at the problem from my book: x²+4/X²-3x
and there are NO parentheses.

I will ask my teacher today in class why this is.

Does your book show

\(\displaystyle \dfrac{x^2 + 4}{x^2 - 3x}\ or\ x^2 + 4 / x^2 - 3x?\)

If your book shows the second expression to mean the first, then your book is wrong. \(\displaystyle x^2 + 4 / x^2 - 3x = \dfrac{x^4 - 3x^3 + 4}{x^2}.\)

This is PEMDAS, which presumably you learned in pre-algebra.

The source of your confusion, I suspect, is that the bar in a VERTICAL fraction acts as a grouping symbol for both numerator and denominator.

 

[HallsofIvy]

Hello!

I was taking a math quiz yesterday and came across this problem:
It said, "For what value or values is the rational expression undefined?"
Then x +2/x+5.

I am familiar with rational expressions. You have to make x equal to zero to show that anything divided by zero is undefined.
And then you can choose which side will let you do this. Usually it is the denominator. Right?

But I am used to seeing something like x² + 4 / x² - 3x

Really? So you are NOT used to seeing parentheses? Do you understand that what you wrote is \(\displaystyle x^2+ (4/x^2)- 3x\) and is undefined only for x= 0? IF the problem were \(\displaystyle (x^2+ 4)/(x^2- 3x)\) THEN it would be undefined for x= 0 or x= 3.

So I immediately know to take the denominator expressions of x² - 3x
and say x(x+3)
Then x-3=0
X-3
And x=0

X is undefined for 0,3.

BUT, when I saw the question on the test for x-2/x-5

I assume you mean (x- 2)/(x- 5)

it wasn't as clear to me which values would make it undefined because they both look the same.
And I only got one value from either.

What does "both" or "either" refer to?

Like x-5=0
X=5

So is x only undefined for one value?: 5

No, "x" is defined for all values! It is the value of the fraction that is undefined when the denominator is 0.
And that happens only for x= 5, of course.

And if so why not say x-2=0
X=2

Did you do the arithmetic? If x= 2, (x- 2)/(x- 5)= (2- 2)(2- 5)= 0/(-3)= -0/3. What is tht equal to?
And do you understand what "undefined" means.

I just wasn't sure.

On my quiz, I answered "X is undefined for 0,5."

But I regret putting the zero because that was definitely wrong.

Thanks for any help!

Think about what "undefined" means and why (x- 2)/(x- 5) is "undefined" when x= 5.

 

[Steph Annie]

Dear JeffM,

Yes, it is the first one that you wrote. The "ratio" of two polynomials.
It looks just like a fraction, only with polynomials.

Is there a key function you used to represent that?

I couldn't find it so that is why I wrote "/" to represent a fraction bar.

Thank you!

Does your book show

\(\displaystyle \dfrac{x^2 + 4}{x^2 - 3x}\ or\ x^2 + 4 / x^2 - 3x?\)

If your book shows the second expression to mean the first, then your book is wrong. \(\displaystyle x^2 + 4 / x^2 - 3x = \dfrac{x^4 - 3x^3 + 4}{x^2}.\)

This is PEMDAS, which presumably you learned in pre-algebra.

The source of your confusion, I suspect, is that the bar in a VERTICAL fraction acts as a grouping symbol for both numerator and denominator.

 

[DrPhil]

Dear JeffM,

Yes, it is the first one that you wrote. The "ratio" of two polynomials.
It looks just like a fraction, only with polynomials.

Is there a key function you used to represent that?

I couldn't find it so that is why I wrote "/" to represent a fraction bar.

Thank you!

The "/" is perfectly fine, but it does require "extra" parentheses when typing inline, so that the order of operations is unambiguous.

If you want to take the time to learn LaTeX, the code to make a display fraction is "\display{..}{..}". For instance:
\dfrac{x^2 + 4}{x^2 - 3x}

Taking that same code and enclosing it in tags \(\displaystyle \text {\(\displaystyle }\) and \(\displaystyle \text{\)}\) gives

\(\displaystyle \dfrac{x^2 + 4}{x^2 - 3x}\)

 

[JeffM]

Dear JeffM,

Yes, it is the first one that you wrote. The "ratio" of two polynomials.
It looks just like a fraction, only with polynomials.

Is there a key function you used to represent that? Unfortunately, no.

I couldn't find it so that is why I wrote "/" to represent a fraction bar. There is nothing wrong with using / to represent a fraction, and doing so is more convenient when using a keyboard. If you use a / to write fractions horizontally, however, you have to be quite careful of grouping symbols and PEMDAS.

Thank you!

Some of us who answer lots of question believe that our answers are easier to understand if we use a feature of this site called LaTeX. It is a somewhat fussy scripting language. Unless you plan on asking lots of questions, I do not recommend trying to learn it: your primary task is to learn what are required topics in school, not learn a scripting language useful for mathematics.