A rational expression is a fraction where both the numerator and the denominator are polynomials. An example isSometimes the numerator and denominator can be named using function notation. For example, suppose and Rational expressions are undefined for values of that make the denominator equal . Therefore, cannot equal
Write the rational expression in its simplest form.
Since rational expressions are essentially fractions, it's possible to add, subtract, multiply and divide them. When creating a multiple of a rational expression, by multiplying by the factor in both the denominator and the numerator, the equality still holds true.
Canceling out a factor yields the same equality. In both cases, the factor can take all values except
It is also possible to create a multiple or cancel out a factor by using a more complex polynomial: Consider the domain. The first expression is undefined for but the second expression is not. It looks like the domain has been expanded when was canceled out, but this is not the case. For the equality to hold true, all -values must give the same value on both sides. Taking this into account gives
Multiplying rational expressions works the same as multiplying fractions. The numerators and denominators are multiplied separately.
To divide two rational expressions, the first step is to invert the expression in the denominator, and then multiply, similar to dividing fractions.
When rewriting a division of a rational expression as multiplication, it might appear to change the domain. For example, is undefined for the -values and Each of these -values result in the expression in any of the denominators being equal to . The rewritten expressionis, however, undefined only for the -values and In order to be able to have an equality between the expressions they must have the same domain. Therefore, and are equal for all except
Determine the product and quotient of the rational expressions. Simplify completely.
When adding and subtracting rational expressions, the same rules apply as when adding and subtracting fractions. If they share a denominator, the numerators can be added or subtracted directly.
Determine the sum of the rational expressions.