RATIONAL EXP.

Algebraic fractions. Polynomials divided by polynomials. The study of infinite asymptotes and mathematical singularities.

01 // The Singularity

A rational expression is simply a fraction where the numerator and denominator are polynomials. However, because variables live in the denominator, you face a critical danger: Division by Zero.

Domain Restriction

Any x-value that makes the denominator equal to exactly zero is strictly excluded from the domain. The graph physically cannot exist at that point, creating an invisible wall.

02 // Simplifying & Holes

1. Factor

\frac{x^2 - 9}{x + 3}

2. Expand

\frac{(x-3)(x+3)}{(x+3)}

3. Cancel

x - 3

Hole at x = -3

The Cancellation Rule: You can only cancel factors (pieces bound by multiplication). If a restriction gets cancelled out of the denominator, it ceases to be an asymptote and becomes a literal "hole" in the graph.

03 // Arithmetic Operations

Mult & Divide

Multiply straight across the top and bottom. For division, multiply by the reciprocal (Keep-Change-Flip).

\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}

Add & Subtract

Requires a Lowest Common Denominator (LCD). This forces you to multiply the top and bottom of each fraction by the missing factors.

\frac{1}{x} + \frac{1}{y} = \frac{y+x}{xy}

Singularities Mapped

You have mastered polynomial fractions and exclusions.

Next: Radicals