What constitutes a rational expression?

Question

How would you explain this concept to someone unfamiliar with it? Demonstrate with one or more examples.

Explain the basic steps involved in simplifying rational expressions. What makes a rational expression undefined?

How are the operations of multiplication, division, addition, and subtraction of rational expressions similar or different from operations on fractions?

Answer 1

A rational expression is a ratio of two polynomials.  A polynomial is an expression that can have constants, variables and exponents, but:

- is not divided by a variable. [like 2/(x+2)]
- a variable's exponents can only be 0,1,2,3,... etc. (no negatives or fractions)
- it can't have an infinite number of terms.

So an example of a rational expression would be:

(x² + 5) / (x + 2)

It is "Rational" because one polynomial is divided by the other, like a ratio.

Simplifying a rational expression is reducing it just like a regular fraction, to simplest terms. In the case of polynomials that means the least number of terms (a polynomial term is every section of the polynomial separated by + or -). In the above rational expression we can infer that x != -2 (not equal) because the result would make the denominator 0 and that makes it undefined (just like regular fractions).

An example of simplifying a rational expression:

 

x² + 5x + 6

      x + 2

 

First we factor the numerator:

x2 + 5x + 6 = (x + 3)(x + 2)

     x + 2               x + 2

 

See the x + 2 term on both the top and bottom? They cancel each other out leaving:

x + 3

As the simplified answer.

As for operations in rational expressions they function exactly as they do in fractions EXCEPT you can not break apart terms linked by a + or - sign (this is why you have to factor). You can combine like terms, you can factor larger polynomials ( 2nd, 3rd, 4th degree ect.) but you absolutely can't for instance say:

x + 6   =   3        thinking that x + 6  =  1 + 2  =  3

x + 3                                     x    3

Hope this helps :)