Rational Expressions

A rational expression is simply a fraction with polynomials on the top and bottom, like $\frac{x + 2}{x + 3}$. "Rational" comes from "ratio," the same way a rational number is a ratio of whole numbers. They show up whenever one quantity is divided by another, like rates and combined resistances. The formulas up top are the fraction rules you already know: $\frac{ad + bc}{bd}$ (get a common bottom, then add the tops), $\frac{ac}{bd}$ (multiply straight across) and $\frac ab\cdot\frac dc$ (to divide, flip the second one and multiply). The algebra can look messy, but every step is something you'd do with $\frac34$.

Same rules as number fractions. You simplify $\frac{6}{8}$ by spotting the common factor $2$ and canceling it. A fraction like $\frac{x^2 - 4}{x^2 + x - 6}$ works the same way, except the common factor is hiding until you factor, which is why the last lesson came first:

Adding and subtracting. With numbers you'd rewrite $\frac12 + \frac13$ over $6$. Here you find the least common denominator by factoring each denominator and taking each factor the most times it shows up in any one of them.

Domain. The original expression is undefined wherever any denominator is zero, and that stays true even if the factor cancels. Above, $x = 2$ makes the original bottom zero, so it's still off limits after simplifying, even though $\frac{x + 2}{x + 3}$ looks perfectly happy there.

A common mistake. It's tempting to cancel the $x^2$'s in $\frac{x^2 - 4}{x^2 + x - 6}$ and be done. You can only cancel factors, things multiplied by everything else, never terms that are added. Try it with numbers: $\frac{2 + 3}{2 + 5} = \frac58$, and crossing out the $2$'s would give $\frac35$.

Lens equations, combined resistances ($\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$) and average speed over round trips are rational expressions.