A rational expression is just a fraction with polynomials in the numerator and denominator. The algebra is usually easy; the discipline is remembering that division by zero is never allowed, which restricts the domain and can silently invalidate a solution you find later.
Domain first, always
Before doing anything else, find every value that makes any denominator zero — those values are excluded from the domain, permanently, no matter what algebra happens afterward.
- excludes
- excludes
- excludes (factor the denominator first)
Solving rational equations
The standard method: factor everything, note the excluded values, clear denominators by multiplying both sides by the least common denominator, solve the resulting polynomial equation, then check every solution against the excluded values. A solution that happens to equal an excluded value is called extraneous — it must be discarded even though it satisfies the cleared-denominator equation.
Worked example
Problem: Solve .
Solution: The domain excludes . Multiply both sides by:
So . But is exactly the excluded value — it's extraneous. This equation has no solution.
Practice problems
1.State the domain restriction(s) for .
2.Simplify .
3.Solve .
4.Solve and note anything unusual.