Integrated Math 3 / Unit 2: Rational Expressions and Functions
2.1 Simplifying Rational Expressions
Notes
A rational expression is a polynomial divided by a polynomial. To simplify, factor the top and bottom and cancel common factors (never common terms).
- $\displaystyle \frac{x^2 - 9}{x^2 + x - 6} = \frac{(x - 3)(x + 3)}{(x + 3)(x - 2)} = \frac{x - 3}{x - 2}$.
- Excluded values make the original denominator 0. Here $x \neq -3$ and $x \neq 2$.
- You cannot cancel terms: $\displaystyle \frac{x + 4}{4}$ does not simplify to $x + 1$.
Practice
Simplify $\displaystyle \frac{x^2 + 5x + 6}{x^2 + 3x + 2}$.
Factor: $\displaystyle \frac{(x + 2)(x + 3)}{(x + 1)(x + 2)}$ and cancel $x + 2$.
Which values are excluded from the domain of $\displaystyle \frac{x + 1}{x^2 - 5x + 6}$?
The denominator factors as $(x - 2)(x - 3)$, which is 0 at $x = 2$ and $x = 3$.
Which statement about $\displaystyle \frac{x + 4}{4}$ is correct?
The 4 in the numerator is a term, not a factor, so it cannot be canceled.
Ready for more? Take the Unit 2 test.