Rational Expressions and Equations
What you must understand
A rational expression is a quotient of polynomials. Simplifying means factoring numerator and denominator completely and canceling common factors, not canceling individual terms. Adding or subtracting requires a common denominator, found from the least common multiple of the factors. Multiplying is straightforward (factor and cancel across), while dividing requires multiplying by the reciprocal. Rational equations are solved by multiplying both sides by the least common denominator to clear fractions, then solving the resulting polynomial equation. Because that step can introduce extraneous solutions, every candidate answer must be checked against the original denominators. The domain of a rational expression excludes any value making a denominator zero.
Traps to watch
- Canceling a term rather than a factor, e.g. treating (x + 3)/3 as x.
- Forgetting to list restricted values before clearing denominators.
- Losing a sign when subtracting rational expressions with a common denominator.
- Failing to check for extraneous solutions after multiplying by the LCD.
- Adding denominators instead of finding a common one.
Worked example
Solve 1/x + 1/(x - 2) = 3/(x(x - 2)). The LCD is x(x - 2), with restrictions x not equal to 0 or 2. Multiply through: (x - 2) + x = 3, so 2x - 2 = 3, x = 2.5. Since 2.5 is not a restricted value, it is valid.
Rule of thumb
Factor, note restrictions, then clear denominators; finish by checking each answer against the original denominators.