Simplifying

You have reduced fractions like $\dfrac{6}{8} = \dfrac{3}{4}$ for years. A rational expression is exactly the same idea, but the top and bottom are polynomials instead of plain numbers — for example $\dfrac{x^2 - 4}{x - 2}$. Every skill you already have for fractions (reduce, multiply, add) carries straight over; we just factor with $x$'s now instead of numbers. Nothing here is brand new — it is fraction arithmetic wearing algebra clothes.

There is one thing numbers-only fractions never forced you to think about: a denominator can accidentally become zero, and division by zero is undefined. The values of the variable that would make the denominator zero are called non-permissible values (or restrictions), and they must be excluded. For $\dfrac{x + 1}{x - 3}$, the value $x = 3$ makes the bottom zero, so we write the restriction $x \ne 3$. Always spot the restrictions before you simplify.

To reduce $\dfrac{6}{8}$ you factored ($\dfrac{2 \cdot 3}{2 \cdot 4}$) and cancelled the common $2$. Rational expressions work the same way: factor the top and bottom completely, then cancel any factor they share. For $\dfrac{x^2 - 4}{x - 2}$, factor the top as a difference of squares: $\dfrac{(x - 2)(x + 2)}{x - 2}$. The $(x - 2)$ appears top and bottom, so it cancels, leaving $x + 2$. Factoring is the whole game — once it is factored, cancelling is easy.

This is the single most common mistake in the entire chapter, so tattoo it on your brain: you can only cancel things that are multiplied (factors), never things that are added or subtracted (terms). In $\dfrac{x + 5}{x + 3}$ you cannot 'cancel the $x$'s' — the $x$ is a term added to a number, not a factor. It's like claiming $\dfrac{2 + 5}{2 + 3} = \dfrac{5}{3}$, which is nonsense ($\dfrac{7}{5} \ne \dfrac{5}{3}$). Cancel only whole factors that multiply the entire top and bottom.

Here is a subtlety worth care. When you cancel a factor like $(x - 2)$, that factor still 'remembers' its restriction. So $\dfrac{(x - 2)(x + 2)}{x - 2} = x + 2$, but with $x \ne 2$ — even though the simplified form $x + 2$ shows no problem at $x = 2$. Read the restrictions off the original denominator (before cancelling), because the simplified expression can hide values that were never allowed.

Simplifying is only as good as your factoring, so keep these patterns sharp: pull out a common factor ($2x + 6 = 2(x + 3)$); the difference of squares ($x^2 - 9 = (x - 3)(x + 3)$); and trinomials ($x^2 - x - 6 = (x - 3)(x + 2)$). Factor the numerator and denominator fully with these tools first — the common factors, and therefore the answer, then jump right out.