Rational Expressions

Half a pizza is $\frac{1}{2}$ , and two slices of $\frac{1}{4}$ also make a half — you have owned the intuition of fractions for years. Now swap the numbers in the numerator and denominator for expressions, say $\frac{3}{x}$ or $\frac{x+1}{x-3}$ : that is a rational expression. Relax — not a single rule the pizza taught you is repealed.

A rational expression is algebra wearing a fraction's coat

A fraction divides a number by a number; a rational expression divides an expression by an expression: whenever numerator or denominator holds a polynomial, it counts. The old routines — cancelling, common denominators, adding like fractions — were practiced in fraction arithmetic, and they transfer unchanged, with expressions standing in for numbers.

InteractiveFraction Pizza Comparator

3/4

Numerator

Denominator

3/4 > 5/8

A greater than B

5/8

Numerator

Denominator

Compare with: 3/4 = 6/8 —— Equivalent multiplier ×2

Warm up on the pizza first: which is more, $\frac{3}{6}$ or $\frac{1}{2}$ ? Laid side by side they are equal — because 3 and 6 were divided by the same 3. That "divide both" move is the soul of cancelling, up next.

Cancelling: common factors leave together

The basic property of fractions: multiply or divide numerator and denominator by the same nonzero number and the fraction keeps its value. Rational expressions copy it verbatim: multiply numerator and denominator by the same nonzero expression and nothing changes. So

$\frac{6x}{9} = \frac{2x}{3}, \qquad \frac{x^2 + 3x}{x} = \frac{x(x+3)}{x} = x + 3$

The key move in the second one is factorising: split numerator and denominator into products first, so that common factors have something to cancel — the same gesture as cancelling the 3 in $\frac{3}{6}$ .

Cancel factors, never addends

$\frac{x+2}{2} \neq x + 1$ ! The numerator is a sum held together by plus signs; the 2 is one addend among two, not a factor. Substitute $x = 2$ to check: the left side is $\frac{4}{2} = 2$ , the right side is 3 — caught red-handed. Cancelling only works on factors of a product.

The denominator may not be zero

The fraction $\frac{3}{0}$ means nothing, and rational expressions inherit the law: the instant a denominator becomes zero, the whole expression breaks. So every rational expression carries a hidden rule — $\frac{3}{x}$ requires $x \neq 0$ , and $\frac{x-2}{x-5}$ requires $x \neq 5$ . A zero numerator is harmless: $\frac{0}{5} = 0$ is perfectly legal, just worth zero.

The red line travels with the expression: $\frac{x^2-1}{x+1} = \frac{(x+1)(x-1)}{x+1} = x - 1$ , but the condition $x \neq -1$ must be carried along — the original denominator was not allowed to be zero, and the simplified form obeys the same rule.

Simplify, then verify with a number

Simplify $\frac{x^2 - 4}{x^2 + 4x + 4}$ . The numerator is the difference of squares $(x+2)(x-2)$ and the denominator is the perfect square $(x+2)^2$ — special products running in reverse. Cancel $(x+2)$ to get $\frac{x-2}{x+2}$ , with $x \neq -2$ . Verify: at $x = 1$ the original is $\frac{-3}{9} = -\frac{1}{3}$ and the simplified form is $\frac{-1}{3}$ — they agree, so the simplification holds.

Substitute a number, the final insurance

Is the simplification right? Substitute and see: pick an $x$ not on the banned list, and the expression before and after cancelling must produce the same value. The trick is an old friend from substitution. Build the three-step habit: write down the denominator's condition, factorise and cancel, then substitute a number to check — and rational expressions will never trip you again.

Check yourself

Quick quiz

1. What is x/3 + x/3?
2. What is (x² − 1)/(x + 1) simplified?
3. For which x is (x + 1)/(x − 3) meaningless?