Simplify rational expressions by factoring each polynomial to its simplest terms, then cancelling out like terms in the numerator and denominator. It may be necessary to multiply by one of the variables to get rid of complex fractions or roots.
As a simple example, take the rational expression (x^2 + 3x + 2)/(x +4). To factor this expression, bring it to its simplest terms by thinking, "What factors of 2 can add up to 3?" In this case, 1 and 2 are the factors, so break it down to [(x + 1)(x + 2)]/(x+ 4). Had the denominator been (x+1), it would have canceled out one of the terms in the numerator, leaving the expression as (x + 2).
Rational expressions and polynomials can also be factored by grouping. Arrange the terms from highest exponent to lowest, then set parentheses around groups of two terms. Bring out common factors, and add them into their own group. In the expression (x^3 + x^2 + 3x + 3), separate them into (x^3 + x^2) + (3x +3). This turns into x^2(x + 1) + 3(x + 1). This in turn leads to (x^2 + 3)(x + 1).
Remember also that the denominator can never equal zero.