What are rational expressions? Why must we always be mindful of the final value of the denominator in a rational expression?  For example, consider the rational expression 3x(x^2-16).  What value...

What are rational expressions? Why must we always be mindful of the final value of the denominator in a rational expression?  For example, consider the rational expression 3x(x^2-16).  What value in the denominator must we be mindful of?

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degeneratecircle's profile pic

degeneratecircle | High School Teacher | (Level 2) Associate Educator

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Every source I can find (after an admittedly quick search) defines "rational expression" in such a way that any polynomial is a rational expression. According to these definitions, `3x(x^2-16)` is a rational expression, and considering the analogy between rational numbers and rational expressions, it should be.


However, that's just a tangential point that hinges on a definition that may not be standard. I agree with the main idea that smallsterea meant to type

`(3x)/(x^2-16)` , and so we need to be aware of the points `x=+-4.`

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crmhaske's profile pic

crmhaske | College Teacher | (Level 3) Associate Educator

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A rational function and a rational expression are not equivalent.  A rational function can have as you have described Q(x)=1, but a rational expression requires both the numerator and denominator to be of at least the degree 1.  The student was asking about rational expressions, not rational functions.  All rational expressions are rational functions, but not all rational functions are rational expressions.

crmhaske's profile pic

crmhaske | College Teacher | (Level 3) Associate Educator

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Correction to the improperly displayed function:

`f(x)=(P(x))/(Q(x))` 

crmhaske's profile pic

crmhaske | College Teacher | (Level 3) Associate Educator

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A rational expression is one that is defined by the ratio of two polynomials, as such:

`f(x)=P(x)/Q(x)`

We can change your function into a rational function by making one of the polynomials the denominator:

`P(x)=3x and Q(x)=x^2-16`

`f(x)=(3x)/(x^2-16)`

It is important to be mindful of the denominator because it cannot be equal to zero.  Any number divided by zero is infinity, and not a real number.  This means that at that point in the graph the function will be undefined.  For this particular expression, it is undefined at `x=+-4` .  Note that in the graph of the function as x approaches 4 or -4, it converges twoards `y=+-oo`

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pramodpandey's profile pic

pramodpandey | College Teacher | (Level 3) Valedictorian

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The expression that  included in  question is  a rational expression.  A concept of  rational expression  is  derived from concept of rational number. A role played by polynomial in rational expression is same what role played by an integer in rational number. All integers are rational number . Thus can we say all polynomials are rational expression ?  Ans. to this question is an affirmative yes.

Let P(x) is polynomial.

If we write `(P(x))/1`  then it is a rational expression.

What change here we observed ! A polynomial is always rational expression.

Here in ref. problem ,he is talking about denominator , may possible he forgot or we donot know exactly what it was .But what question we have ,we need to address it

If we imagine

`(3x(x^2-16))/1`  then denominator is 1.

So here we have answer to his question " What value in the denominator must we be mindful of? " . Answer is 1. 

His second question " Why must we always be mindful of the final value of the denominator in a reational expression?" .

In real number system any number divided by zero is infinity, and not a real number.Thus the denominator of rational expression can not be zero same as in  rational number.Here we are not concern about domain and range etc..  

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