Mathematical formulas

Algebra

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1Numbers and algebraic properties
2Powers, exponents, and radicals
3Algebraic expressions
4Notable products and identities
5Factorization
6Rational expressions and partial fractions
7Equations
8Inequalities
9Systems of equations
10Functions
11Polynomials
12Exponentials and logarithms
13Complex numbers
14Sequences, finite series, and recurrences
15Vectors
16Matrices
17Determinants and inverses
18Vector spaces
19Linear transformations
20Eigenvalues and diagonalization
21Orthogonality and projections
22Least squares
23Matrix decompositions
24Matrix norms and conditioning
25Boolean algebra
26Modular arithmetic
27Algebraic structures
28Linear codes and error correction
29Relationship maps

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Algebra
Rational expressions and partial fractions

Section 6

Rational expressions and partial fractions

Domain of a rational expression

fundamental

R(x)=\frac{P(x)}{Q(x)},\quad Q(x)\ne0

The denominator cannot be zero.

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Sum of algebraic fractions

fundamental

\frac ab+\frac cd=\frac{ad+bc}{bd}

Addition via a common denominator.

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Product and division of fractions

fundamental

\frac ab\frac cd=\frac{ac}{bd};\quad \frac{a/b}{c/d}=\frac{ad}{bc}

Rules for products and quotients of algebraic fractions.

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Partial fractions: distinct linear factors

intermedio

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

Decomposes a rational function into simple terms.

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Partial fractions: repeated factor

intermedio

\frac{P(x)}{(x-a)^n}=\sum_{k=1}^{n}\frac{A_k}{(x-a)^k}

Decomposition for repeated linear factors.

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Partial fractions: irreducible quadratic

intermedio

\frac{Ax+B}{x^2+px+q}

Form associated with an irreducible quadratic factor.

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