A function of the form
$$ f(x) = \frac{P(x)}{Q(x)} $$
where $P(x)$ and $Q(x)\neq 0$ are polynomials is called a rational function. More on rational functions is given in Chapter 3. For now, we will refer to them as rational expressions.
Addition/Subtraction
When adding or subtracting two rational expressions, use the steps below.
Fully factor all denominators.
Determine the common denominator consisting of the fewest factors.
When all rational expressions have the same denominator, add/subtract the numerators while keeping the common denominator.
Leave your denominator in fully factored form.
Check your numerator to see if it factors. Cancel any common factors and note any restrictions.
Adding/subtracting rational expressions, part 1.
Adding/subtracting rational expressions, part 2.
Multiplication/Division
When multiplying two rational expressions, use the steps below.
Fully factor all numerators and denominators.
Cancel any common factors and note any restrictions.
Multiply the numerators and denominators.
Leave your solution in fully factored form.
Multiplying rational expressions.
When dividing two rational expressions, use the steps below.
Fully factor all numerators and denominators.
Multiply the first rational expression by the reciprocal of the second rational expression.
Cancel any common factors and note any restrictions.
Leave your solution in fully factored form.
Dividing rational expressions.
Simplifying Rational Expressions
The video below gives more examples of simplifying difficult rational expressions.
Simplifying rational expressions.