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 :)