Rational fractions

[Vantage]

Rational fractions are defined on Wikipedia as "an algebraic fraction such that both the numerator and denominator are polynomials", or $$f(x) = \frac {P(x)} {Q(x)}$$ How is it then that $$\frac {3} {x-3}$$ is considered a rational fraction? 3 is not a polynomial, nor is x-3. Is it that P(x) is equal to 3, but P(x) itself is a polynomial? I'm not sure I just don't understand it, or either I'm making a mistake in my understanding of polynomials.

Thanks in advance for any help.

 

[Deleted member 4993]

Rational fractions are defined on Wikipedia as "an algebraic fraction such that both the numerator and denominator are polynomials", or $$f(x) = \frac {P(x)} {Q(x)}$$ How is it then that $$\frac {3} {x-3}$$ is considered a rational fraction? 3 is not a polynomial, nor is x-3. Is it that P(x) is equal to 3, but P(x) itself is a polynomial? I'm not sure I just don't understand it, or either I'm making a mistake in my understanding of polynomials.

Thanks in advance for any help.

Definition of polynomial (according Meriam-Webster):

a mathematical expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variables raised to a nonnegative integral power (such as a + bx + cx²)

Thus a constant (such as 3) is actually = 3*x⁰ → thus a polynomial

 

[Otis]

Rational fractions are defined on Wikipedia as "an algebraic fraction...

Hi Vantage. Did you intend to write "rational functions" above? All fractions represent Rational numbers, regardless of whether they contain algebraic expressions.

The expression x-3 is a linear polynomial.

Here is a link to a brief introduction to polynomials.

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[JeffM]

Hi Vantage. Did you intend to write "rational functions" above? All fractions represent Rational numbers, regardless of whether they contain algebraic expressions.

The expression x-3 is a linear polynomial.

Here is a link to a brief introduction to polynomials.

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$\dfrac{\sqrt{2}}{\pi}$ is a rational number? Who knew?

 

[Vantage]

Definition of polynomial (according Meriam-Webster):

a mathematical expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variables raised to a nonnegative integral power (such as a + bx + cx²)

Thus a constant (such as 3) is actually = 3*x⁰ → thus a polynomial

Thank you ever so much. I was under the impression that polynomials had to include a variable with an exponent of minimum 2. I don't know where I got that from, but now I know it's very wrong my understanding is much clearer of rational functions and fractions.

 

[Dr.Peterson]

Rational fractions are defined on Wikipedia as "an algebraic fraction such that both the numerator and denominator are polynomials", or $$f(x) = \frac {P(x)} {Q(x)}$$ How is it then that $$\frac {3} {x-3}$$ is considered a rational fraction? 3 is not a polynomial, nor is x-3. Is it that P(x) is equal to 3, but P(x) itself is a polynomial? I'm not sure I just don't understand it, or either I'm making a mistake in my understanding of polynomials.

Thanks in advance for any help.

Did you intend to write "rational functions" above?

The quote is from the article on rational function, and uses the term "rational fraction" in defining it.

We can't be sure of @Vantage's reason for saying "3 is not a polynomial, nor is x-3", without further input. Many textbooks make a point of giving examples showing that 3, x-3, and so on, are in fact polynomials, because students often get the wrong impression that there must be more than one term, or it must include the variable, or there must be an explicit exponent, or even (from the meaning of "poly" as "many") that there must be more than two terms. (The link in the Wikipedia article to Polynomials, unfortunately, doesn't emphasize this, though it supports it by not mentioning the number of terms at all.)

 

[Vantage]

The quote is from the article on rational function, and uses the term "rational fraction" in defining it.

We can't be sure of @Vantage's reason for saying "3 is not a polynomial, nor is x-3", without further input. Many textbooks make a point of giving examples showing that 3, x-3, and so on, are in fact polynomials, because students often get the wrong impression that there must be more than one term, or it must include the variable, or there must be an explicit exponent, or even (from the meaning of "poly" as "many") that there must be more than two terms. (The link in the Wikipedia article to Polynomials, unfortunately, doesn't emphasize this, though it supports it by not mentioning the number of terms at all.)

This is correct. I thought I had a decent understanding of polynomials, but I still checked Wikipedia for the definition. I misinterpreted the format and thought that all polynomials had to have a variable with a minimum index of 2, which is incorrect. Thank you for the clarification.

 

[Deleted member 4993]

Thank you ever so much. I was under the impression that polynomials had to include a variable with an exponent of minimum 2. I don't know where I got that from, but now I know it's very wrong my understanding is much clearer of rational functions and fractions.

Most important term to notice in that definition is:

nonnegative integral power​

Thus expressions containing exponents of x such as ¼ or ¾ or π or √3, etc. will NOT be considered a polynomial.

 

[JeffM]

This is correct. I thought I had a decent understanding of polynomials, but I still checked Wikipedia for the definition. I misinterpreted the format and thought that all polynomials had to have a variable with a minimum index of 2, which is incorrect. Thank you for the clarification.

Wikipedia articles on mathematics are often pitched at a fairly high degree of mathematical sophistication. Purple math is often a good source to check out if the Wiki article seems to be pretty deep. (A problem with Purple Math is that it is not always easy to find things.) Here is its definition of a polynomial

Polynomials: Their Terms, Names, and Rules Explained

What is a polynomial? This lesson explains what they are, how to find their degrees, and how to evaluate them.

www.purplemath.com

It initially gives a definition that is not technically advanced, but, toward the end, it discusses constants as a numeral times the variable to the zero power.

Purple math does not have an entry for rational expressions, but it does have entries concerning rational expressions and equations. If you look at those, one starts with a definition of a rational expression.

https://www.purplemath.com/modules/rtnldefs.htm

And of course, you can always come here if you are uncertain of what a definition means.

 

[Vantage]

Wikipedia articles on mathematics are often pitched at a fairly high degree of mathematical sophistication. Purple math is often a good source to check out if the Wiki article seems to be pretty deep. (A problem with Purple Math is that it is not always easy to find things.) Here is its definition of a polynomial

Polynomials: Their Terms, Names, and Rules Explained

What is a polynomial? This lesson explains what they are, how to find their degrees, and how to evaluate them.

www.purplemath.com

It initially gives a definition that is not technically advanced, but, toward the end, it discusses constants as a numeral times the variable to the zero power.

Purple math does not have an entry for rational expressions, but it does have entries concerning rational expressions and equations. If you look at those, one starts with a definition of a rational expression.

https://www.purplemath.com/modules/rtnldefs.htm

And of course, you can always come here if you are uncertain of what a definition means.

Ah, that's excellent. That website seems invaluable for what I'm studying. Many thanks for all the help

 

[Otis]

$\dfrac{\sqrt{2}}{\pi}$ is a rational number?

I did stumble in wording my thoughts, ha. The phrase "rational fraction" threw me. Never seen it.

Wiki defines Rational functions in terms of rational fractions, which they say are algebraic fractions, which are then a ratio of polynomials? Good grief!

 

[JeffM]

I did stumble in wording my thoughts, ha. The phrase "rational fraction" threw me. Never seen it.

Wiki defines Rational functions in terms of rational fractions, which they say are algebraic fractions, which are then a ratio of polynomials? Good grief!

Indeed.

I love wikipedia. I give it a nice donation every year. But it has to be used with great care.

 

[Otis]

wikipedia...has to be used with great care.

Agree! Going around the maypole made it seem like sqrt(x)/x is not an algebraic fraction, until my head stopped spinning.

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