Question

How to simplify algebraic fractions?

Answer 1

There is no single method for simplifying algebraic fractions.

Explanation:

Some algebraic fractions can not be simplified.
For example: `x/(y+1)` can not be simplified.

Sometimes a common factor can be found in the numerator and denominator and removed.
For example: `(x^2-1)/(xy-2x-y-2)=((x+1)(x-1))/((x-1)(y+2))=(x+1)/(y+2)`

Sometimes you can perform polynomial long division and find a simplification; but sometimes whether the result is a "simplification" is questionable:
For example: `(2x^3+4x^2+9)/(x-3)`
if treated as a long division gives the result:
`color(white)("XXX")2x^2+5x+15 +54/(x-3)`
Is this really a simplification?