Rational Expressions

A rational expression is a fraction whose numerator and denominator are polynomials. Like fractions of numbers, they simplify by cancelling common factors — never common terms.

Domain restrictions

The denominator can’t be zero. Always identify the values that would zero it out and exclude them from the domain.

State domain restrictions before simplifying. After simplifying, the cancelled factors still create restrictions.

Simplifying

1. Factor numerator and denominator completely.
2. Cancel common factors (entire parenthesized expressions, not individual terms).
3. State domain restrictions for any cancelled factors.

The cancelled $(x+3)$ doesn’t appear in the simplified expression but its restriction $x\ne -3$ survives.

The cancellation rule (THE most common error)

You can cancel factors (multiplied) but never terms (added or subtracted).

• Wrong: $\frac{x+1}{x+2}=\frac{1}{2}$ — the $x$’s aren’t factors.
• Right: $\frac{2x\cdot x}{3\cdot x}=\frac{2x}{3}$ — the $x$’s are factors.

Memorize this rule. Internalize it. Half the algebra mistakes in calculus trace back here.

Multiplying and dividing

Multiply: factor everything, multiply numerators, multiply denominators, cancel any factors that appear in any numerator and any denominator.

Divide: flip the second fraction (multiplicative inverse) and multiply.

Adding and subtracting (LCD method)

Adding rational expressions follows the same rule as adding numerical fractions: a common denominator is required before the numerators can combine.

1. Factor every denominator.
2. Build the least common denominator (LCD): the product of every distinct factor, each raised to the highest power it appears in any single denominator.
3. Rewrite each fraction over the LCD by multiplying numerator and denominator by the missing factors.
4. Add or subtract the numerators (keeping the LCD), then simplify.

Worked example:

This LCD machinery is a prerequisite for next week’s rational equations: clearing denominators in $\frac{1}{x}+\frac{1}{2}=\frac{3}{4}$ is exactly the same LCD construction, just applied to both sides of an equals sign rather than across a single sum.

ML connection — Bayes’ theorem

Bayes’ theorem is structurally a rational expression:

The denominator $P(B)$ is often hard to compute (it’s an integral over all hypotheses). In many ML algorithms — Naive Bayes, MAP estimation — when the denominator is independent of the parameter we’re optimizing, we can drop it (it doesn’t affect which value of the parameter maximizes the expression). That’s the same “cancel factors that don’t depend on what we care about” intuition you just practiced.