Partial fraction decomposition breaks down complex rational expressions into simpler fractions. This technique is crucial for integrating rational functions and simplifying complex expressions in calculus and algebra.

The method involves splitting a fraction with a complex denominator into a sum of fractions with simpler denominators. This process helps in solving integrals, differential equations, and simplifying algebraic expressions.

Partial Fraction Decomposition

Decomposition of rational expressions

• Partial fraction decomposition breaks down a complex rational expression into a sum of simpler fractions
• Useful for integrating rational functions and simplifying complex expressions
• For a rational expression $\frac{P(x)}{Q(x)}$ where $Q(x)$ is a product of distinct linear factors $(x-a)(x-b)(x-c)$, the decomposition takes the form:
  • $\frac{P(x)}{(x-a)(x-b)(x-c)} = \frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c}$
• To find the values of $A$, $B$, and $C$, multiply both sides by the common denominator $(x-a)(x-b)(x-c)$ and solve the resulting equation
  • Equate coefficients of like terms on both sides or substitute specific values for $x$ (such as $x=a$, $x=b$, and $x=c$)
• Proper factorization of the denominator is crucial for successful decomposition

Partial fractions with repeated factors

• When the denominator contains repeated linear factors, the decomposition includes terms with powers of the repeated factor in the denominator
• For a rational expression with a repeated linear factor$ $(x-a)^n$$, the decomposition includes terms of the form:
  • $\frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \cdots + \frac{A_n}{(x-a)^n}$
• The number of terms in the decomposition for a repeated factor is equal to the power of the factor
• To find the values of the coefficients, multiply both sides by the common denominator and solve the resulting equation
  • Equate coefficients of like terms or substitute values for $x$ (such as $x=a$)

Quadratic factors in partial fractions

• When the denominator contains nonrepeated irreducible quadratic factors, the decomposition includes terms with linear numerators over the quadratic factors
• For a rational expression with a nonrepeated irreducible quadratic factor $ax^2 + bx + c$, the decomposition includes a term of the form:
  • $\frac{Ax + B}{ax^2 + bx + c}$
• The numerator is a linear polynomial $Ax + B$ because the degree of the numerator should be one less than the degree of the denominator
• To find the values of $A$ and $B$, multiply both sides by the common denominator and solve the resulting equation
  • Equate coefficients of like terms or substitute values for $x$
• Quadratic factors with complex roots require special consideration in the decomposition process

Repeated quadratics in decomposition

• When the denominator contains repeated irreducible quadratic factors, the decomposition includes terms with polynomial numerators over powers of the quadratic factors
• For a rational expression with a repeated irreducible quadratic factor $(ax^2 + bx + c)^n$, the decomposition includes terms of the form:
  • $\frac{A_1x + B_1}{ax^2 + bx + c} + \frac{A_2x + B_2}{(ax^2 + bx + c)^2} + \cdots + \frac{A_nx + B_n}{(ax^2 + bx + c)^n}$
• The degree of the numerator polynomials should be one less than the degree of the corresponding denominator term
• To find the values of the coefficients, multiply both sides by the common denominator and solve the resulting equation
  • Equate coefficients of like terms or substitute values for $x$

Applications in calculus and differential equations

• Partial fraction decomposition is a powerful technique used in integration of rational functions
• The method simplifies complex rational expressions, making them easier to integrate
• In solving differential equations, partial fractions can be applied to simplify and solve certain types of equations