11.4 Partial Fractions

Partial fraction decomposition is a powerful tool for simplifying complex rational expressions. It breaks down tricky fractions into simpler ones, making them easier to work with in calculus and beyond.

This technique is super useful for integration and solving differential equations. By mastering partial fractions, you'll be able to tackle more advanced math problems with confidence and ease.

Partial Fraction Decomposition

Decomposition of rational expressions

• Breaks down complex rational expression into sum of simpler fractions
• For nonrepeated linear factors in denominator, decomposition takes form:
  • $\frac{P(x)}{(x-a)(x-b)(x-c)} = \frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c}$
• Find values of $A$, $B$, $C$ by multiplying both sides by common denominator and equating coefficients or evaluating at specific points
• Example: $\frac{2x+1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$
  • Multiply both sides by $(x-1)(x+2)$: $2x+1 = A(x+2) + B(x-1)$
  • Equate coefficients or evaluate at $x=1$ and $x=-2$ to solve for $A$ and $B$
• Useful for simplifying complex fractions into more manageable terms (integration, differential equations)

Partial fractions with repeated factors

• For repeated linear factors in denominator, decomposition includes terms with powers of repeated factor
  • $\frac{P(x)}{(x-a)^n(x-b)} = \frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + ... + \frac{A_n}{(x-a)^n} + \frac{B}{x-b}$
• Find coefficients by equating like terms or evaluating at specific points
• Example: $\frac{x^2+3x+1}{(x-1)^2(x+1)} = \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{x+1}$
  • Multiply both sides by $(x-1)^2(x+1)$ and equate coefficients or evaluate at $x=1$ and $x=-1$ to solve for $A$, $B$, $C$
• Repeated factors arise in applications like partial differential equations and control systems

Quadratic factors in partial fractions

• For nonrepeated irreducible quadratic factors in denominator, decomposition includes terms with linear numerators over quadratic factors
  • $\frac{P(x)}{(ax^2+bx+c)(dx^2+ex+f)} = \frac{Ax+B}{ax^2+bx+c} + \frac{Cx+D}{dx^2+ex+f}$
• Find $A$, $B$, $C$, $D$ by multiplying both sides by common denominator and equating coefficients
• Example: $\frac{2x+1}{(x^2+1)(x^2-4)} = \frac{Ax+B}{x^2+1} + \frac{Cx+D}{x^2-4}$
  • Multiply both sides by $(x^2+1)(x^2-4)$ and equate coefficients to solve for $A$, $B$, $C$, $D$
• Quadratic factors appear when dealing with trigonometric functions (sine, cosine) or complex numbers

Repeated quadratics in decomposition

• For repeated irreducible quadratic factors in denominator, decomposition includes terms with polynomial numerators over powers of quadratic factors
  • $\frac{P(x)}{(ax^2+bx+c)^n} = \frac{A_1x+B_1}{ax^2+bx+c} + \frac{A_2x+B_2}{(ax^2+bx+c)^2} + ... + \frac{A_nx+B_n}{(ax^2+bx+c)^n}$
• Find coefficients by equating like terms
• Example: $\frac{x^3+2x^2+x+1}{(x^2+1)^2} = \frac{Ax+B}{x^2+1} + \frac{Cx+D}{(x^2+1)^2}$
  • Multiply both sides by $(x^2+1)^2$ and equate coefficients to solve for $A$, $B$, $C$, $D$
• Repeated quadratic factors occur in advanced calculus topics (improper integrals, series expansions)

Applications in Advanced Mathematics

• Partial fraction decomposition is crucial in solving complex integration problems
• The technique is often used in solving differential equations, especially those with rational functions
• In linear algebra, partial fractions help in decomposing rational matrix functions
• Partial fractions are useful in evaluating limits of rational functions as x approaches infinity