What a rational expression is
A rational expression is a quotient p(x)/q(x) of two polynomials, such as (x² − x − 6)/(x² − 9). It is defined for every x where the denominator q(x) is not zero. Here x² − 9 = 0 at x = 3 and x = −3, so the expression is not defined there.
Simplify: factorise and cancel
Factorise the numerator and the denominator separately and cancel the factors they share:
(x² − x − 6)/(x² − 9) = (x − 3)(x + 2)/((x − 3)(x + 3)) = (x + 2)/(x + 3), where x ≠ 3 and x ≠ −3.
The condition x ≠ 3 stays after cancelling. The simplified expression (x + 2)/(x + 3) has a value at x = 3, but the original does not, and the two are equal only where both are defined.
Add and subtract
Write both fractions over a common denominator, multiply each up, and add the numerators:
2/(x − 1) + 3/(x + 2) = (2(x + 2) + 3(x − 1))/((x − 1)(x + 2)) = (5x + 1)/((x − 1)(x + 2)).
When subtracting, the minus sign applies to the whole second numerator: 3/(x − 2) − 1/(x + 1) has the numerator 3(x + 1) − (x − 2) = 2x + 5.
Equations
First state the values x may not take, then multiply both sides by the denominators and solve. In 3/(x − 1) = 2/(x + 2), x may not be 1 or −2. Multiplying by (x − 1)(x + 2) gives 3(x + 2) = 2(x − 1), so x = −8. That is none of the excluded values, so x = −8 is the solution. Had the answer been an excluded value, the equation would have had no solution.
Common mistakes
- Cancelling terms instead of factors. In (x + 2)/(x + 3) the x cannot be crossed out: those are terms, and 2/3 is a different expression.
- Dropping the condition. After cancelling, the original denominator still may not be zero.
- Adding the denominators. 2/(x − 1) + 3/(x + 2) is not 5/(2x + 1).
- Forgetting the bracket when subtracting. −(x − 2) is −x + 2, not −x − 2.
See also
Polynomial division, for a numerator of higher degree than the denominator, and the difference of two squares, which factorises x² − 9.