Rational expressions — simplification, operations: Foundations

This page assumes you have seen nothing. We will earn every symbol below before the parent note is allowed to use it. Read top to bottom; each block leans on the one before it.

1. What a fraction actually is

Before any letters, look at plain numbers.

Figure — Rational expressions — simplification, operations

Why the topic needs this: every single rule later — cancelling, multiplying, adding — is just a numeric-fraction rule with polynomials pasted in. If $\frac{6}{8} = \frac{3}{4}$ makes sense to you, the whole chapter is the same move.

2. Variables and what a letter stands for

Why the topic needs this: a rational expression (unlike a plain fraction) has these boxes inside it. Because the box can hold any number, some choices might accidentally make the denominator $0$ — and those are exactly the values we must forbid. See Domain and range for the full language of "allowed inputs."

Recall

What does $x$ represent? ::: An unknown number — a box we will fill in later.

3. Terms vs. factors — the distinction the whole topic lives or dies on

This is the single most important idea on the page. Get it wrong and every cancellation goes wrong.

Look at the figure. On the left, $x$ and $3$ sit side by side, added — like two separate tiles. On the right, $x$ and $3$ are glued into one block by multiplication. Cancelling only works on the glued blocks (factors), never on the side-by-side tiles (terms).

Recall

Can you cancel the $x$ in $\frac{x + 3}{x + 5}$ ? ::: No — $x$ is a term (added), not a factor (multiplied). Cancelling needs factors.

4. Polynomial — the thing on top and bottom

Why the topic needs this: this is the object of study. Everything else is machinery for handling it.

5. Factoring — turning a sum back into a product

Cancelling needs factors (§3). But polynomials arrive as sums of terms ( $x^{2} - 4$ ). So we need a machine that rewrites a sum as a product. That machine is factoring.

Why this exact tool and not another? We want to cancel, and cancelling only touches factors. Factoring is the only operation that converts un-cancellable sums into cancellable products. Without it, no simplification is possible. The full menu of techniques (common factor, difference of squares, splitting the middle term) lives at Polynomial factoring.

Recall

Factor $x^{2} - 4$ . ::: $(x - 2) (x + 2)$ — difference of squares with $a = x, b = 2$ .

6. Cancelling — the payoff

Once top and bottom are both products, we may delete a factor that appears in both.

The catch that never leaves: even after $C$ vanishes from the written form, the original expression was undefined wherever $C = 0$ . That forbidden value must still be reported. This is Common Mistake 2 in the parent — the disappearing restriction.

7. The symbols that show up in the rules

Symbol	Plain words	Picture
$\neq =$	"is not equal to"	an $=$ sign with a slash — a road blocked off
$\Rightarrow$	"therefore / this forces"	an arrow: one fact pushes to the next
$\cdot$	multiply	the glue that makes factors (§3)
$\div$	divide	asks "how many of this fit into that?"
LCD	Least Common Denominator	the smallest shared number of pizza slices so two fractions can be added

8. How it all fits together

Read the map top-down: fractions and variables are the ground floor; the term/factor split plus factoring feed cancelling; the zero rule feeds restrictions; both streams merge into the five operations that the parent note drills. Downstream this feeds Complex fractions, Rational equations, Polynomial long division, Partial fraction decomposition, and eventually Limits and continuity (those "holes" from cancelled factors).

Equipment checklist

Cover the right side and test yourself before opening the parent note.

What does the bar in $\frac{a}{b}$ mean? ::: Divide $a$ by $b$ ; the whole is cut into $b$ equal parts and we take $a$ .
Why can the denominator never equal zero? ::: You cannot cut a whole into zero pieces — division by zero has no meaning.
What is a variable like $x$ ? ::: A box holding an as-yet-unchosen number.
Difference between a term and a factor? ::: A term is joined by $+$ or $-$ ; a factor is joined by multiplication.
Which can you cancel — terms or factors? ::: Only factors (things multiplied), never terms (things added).
What is a polynomial? ::: A sum of terms, each a number times a whole-number power of $x$ .
What is a rational expression? ::: A ratio $\frac{P ( x )}{Q ( x )}$ of two polynomials, with $Q (x) \neq = 0$ .
Why do we factor before simplifying? ::: Cancelling needs factors, and factoring turns un-cancellable sums into products.
Factor $x^{2} - 9$ . ::: $(x - 3) (x + 3)$ , difference of squares.
Why is $\frac{A \cdot C}{B \cdot C} = \frac{A}{B}$ true? ::: Because $\frac{C}{C} = 1$ , and multiplying by $1$ changes nothing (needs $C \neq = 0$ ).
After cancelling a factor $C$ , do you still forbid $C = 0$ ? ::: Yes — the original expression was undefined there, so the restriction stays.
Why do fractions need a common denominator to add? ::: The slice-sizes must match before you can combine the pieces on top.