Rewriting Simple Rational Expressions
Rewriting a rational expression means factoring the top and bottom so matching factors cancel, leaving a simpler expression that behaves the same everywhere except at the x-values that made the original denominator zero.
A rational expression is just one polynomial divided by another. To simplify it, factor the numerator and denominator completely, then cancel any factor that appears in both. Only whole factors can cancel this way, not individual terms, because factoring turns the expression into a product, and a product's pieces are what division actually works on. The value of x that makes the cancelled factor zero still breaks the original expression, so that point stays excluded even after the fraction looks simpler. Graphically this shows up as a hole: a single missing point on an otherwise normal curve.
This is also why (x+4)/(x+2) cannot be reduced by crossing out the x's. Nothing is multiplied there, so there is no shared factor to legally remove. Only factored forms, like (x+a)(x-a) over (x+a), give you something real to cancel.
When the denominator does not divide out evenly, you can still rewrite the expression as a polynomial plus a remainder over the original denominator, using polynomial long division or synthetic division. This mirrors how 17/5 becomes 3 plus 2/5. That skill connects directly to graphing rational functions, since the polynomial part reveals end behavior and the remainder term shows how the graph bends near its asymptote. Rewriting rational expressions also sets up adding or subtracting ones with different denominators, which requires a common denominator built from each factored piece, the same way fractions do with whole numbers.
Key terms
- Rational expression
- A fraction whose numerator and denominator are both polynomials.
- Excluded value
- An x-value that makes the original denominator zero, so it cannot be part of the domain.
- Factoring
- Rewriting a polynomial as a product of simpler polynomials.
- Polynomial long division
- A method for dividing one polynomial by another to get a quotient and a remainder.
- Hole
- A single missing point on a graph where a factor cancelled from both numerator and denominator.
Watch it explained
- How it worksHow does factoring the numerator and denominator first guarantee that only whole factors can cancel?
- Worked exampleFind the simplest form of (x2-9)/(x2+x-12) and state the value of x that must be excluded.
- Common mistakeWhy isn't (x+4)/(x+2) equal to 2 after you cancel the x from the top and the bottom?
- Worked exampleShow me how to rewrite (2x2+5x-3)/(x+3) as a polynomial plus a remainder over x+3.
- Going furtherWhat changes when a rational expression has to be added to another one with a different denominator?
Check yourself
End of topic.