A rational expression is just a fraction with polynomials in the numerator and denominator, like (x2 − 9) / (x + 3). It behaves exactly like a number fraction — you can simplify it, multiply it, divide it, and add it — the only twist is that “simplifying” usually means factoring first, then canceling.
Simplifying — factor first, then cancel
You can’t cancel terms that are just sitting next to each other in a sum — only matching factors cancel. So before canceling anything, factor the top and bottom completely. Take (x2 − 9) / (x + 3): the numerator factors as a difference of squares, (x + 3)(x − 3). Now the (x + 3) on top matches the (x + 3) on the bottom and cancels, leaving just x − 3.
Multiplying and dividing — same rules as number fractions
Multiplying rational expressions works exactly like multiplying number fractions: multiply tops together, multiply bottoms together, then factor and cancel anything that matches. Dividing is the same flip-and-multiply move you already know — a/b ÷ c/d = a/b × d/c — just with polynomials standing in for the numbers.
Let me show you one — adding
Add 1/x + 1/(x+2). Just like with number fractions, you need a common denominator first — here, that’s x(x+2). Rewrite each piece: (x+2)/[x(x+2)] + x/[x(x+2)]. Now the denominators match, so add the numerators: (2x + 2) / [x(x+2)]. The process is identical to adding 1/3 + 1/5 — find a common denominator, rewrite each fraction, then combine — only the “numbers” happen to have an x in them.
Restricted values — the one new thing
Because these are fractions, the denominator can never equal zero. In (x2 − 9) / (x + 3), x cannot equal −3 — even after simplifying down to x − 3, that restriction still quietly applies, because it came from the original denominator. Always note the restricted values before you simplify, while the original denominator is still in front of you.
The mistake I see most
Canceling an x that appears in both the numerator and denominator as a stray term, instead of as a shared factor — for example, wrongly canceling the x’s in (x + 2)/x to get just “+2.” The fix is the same rule from the top of this page: only fully factored, matching factors are allowed to cancel, never a piece of a sum.
Want to work through it together?
If rational expressions feel like a brand-new topic, that’s usually a sign the factoring underneath needs a refresh, not that the fraction rules themselves are the problem — and that’s a fast fix with the right worked examples.
