Section 6

Rational expressions and partial fractions

fundamental

R (x) = \frac{P ( x )}{Q ( x )}, Q (x) \neq = 0

The denominator cannot be zero.

fundamental

\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}

Addition via a common denominator.

fundamental

\frac{a}{b} \frac{c}{d} = \frac{a c}{b d}; \frac{a / b}{c / d} = \frac{a d}{b c}

Rules for products and quotients of algebraic fractions.

intermediate

\frac{P ( x )}{( x - a ) ( x - b )} = \frac{A}{x - a} + \frac{B}{x - b}

Decomposes a rational function into simple terms.

intermediate

\frac{P ( x )}{( x - a ) ^{n}} = k = 1 \sum n \frac{A _{k}}{( x - a ) ^{k}}

Decomposition for repeated linear factors.

intermediate

\frac{A x + B}{x ^{2} + p x + q}

Form associated with an irreducible quadratic factor.