algebraic fraction
An algebraic fraction is an ordinary fraction that has grown a variable: instead of a plain number on top or bottom, you find an expression with a letter in it. Just as 3/4 is a number over a number, x/(x + 1) is an expression over an expression.
More precisely, an algebraic fraction is any quotient whose numerator or denominator (or both) contains a variable expression — for example (x + 2)/5, 7/(x - 3), or (x^2 - 1)/(x + 1). The same arithmetic instincts apply: you can simplify by cancelling common factors, and you add fractions over a common denominator.
There is one hard rule the variable cannot break: the denominator must never equal zero, because division by zero is undefined. So in 1/(x - 2), the value x = 2 is forbidden — it is an excluded value. Whenever you work with algebraic fractions, keep an eye on which inputs would make a denominator vanish.
(x^2 - 1)/(x + 1) factors to (x - 1)(x + 1)/(x + 1) = x - 1, valid for all x except x = -1 (where the original denominator is 0).
Cancel common factors, but exclude values that zero the denominator.