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# What constitutes a rational expression?

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?

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:

(x+ 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 x6  =  1 + 2  =  3

x + 3                                     x    3

Hope this helps :)