A **ratio**nal 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^{2 }+ 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:

x^{2} + 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 :)

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