Partial Fraction Decomposition in Precalculus

Rational Functions and Partial Fraction Decomposition

Introduction to Rational Functions

Rational functions are quotients of two polynomials, expressed as f(x) = \( \frac{P(x)}{Q(x)} \), where P(x) and Q(x) are polynomials and Q(x) \neq 0. In calculus and precalculus, it is often useful to rewrite complicated rational functions as sums of simpler fractions, called partial fractions.

• Degree of Numerator (deg(num)): The highest power of x in the numerator.

• Degree of Denominator (deg(den)): The highest power of x in the denominator.

• If deg(num) ≥ deg(den), perform long division first to rewrite the function as a polynomial plus a proper fraction.

Long Division of Polynomials

When the degree of the numerator is greater than or equal to the degree of the denominator, use polynomial long division:

• Divide the numerator by the denominator to obtain a quotient and a remainder.

• The original function can be written as:

$\frac{P(x)}{Q(x)} = S(x) + \frac{R(x)}{Q(x)}$

• Where S(x) is the quotient and R(x) is the remainder (with deg(R) < deg(Q)).

Factoring the Denominator

To decompose a rational function into partial fractions, first factor the denominator into irreducible linear and quadratic terms:

• Every real polynomial can be factored into a product of linear and irreducible quadratic terms.

• Example: $x^2 - 1 = (x - 1)(x + 1)$ (linear factors)

• Example: $x^2 + 1$ is irreducible over the reals (quadratic factor)

Partial Fraction Decomposition: The Method

General Steps

1. Ensure the rational function is proper (deg(num) < deg(den)). If not, use long division first.

2. Factor the denominator completely into linear and irreducible quadratic factors.

3. Set up the decomposition by assigning a separate term for each factor:

• For each linear factor $(x - a)^k$, include terms:

  • $\frac{A_1}{x - a} + \frac{A_2}{(x - a)^2} + \cdots + \frac{A_k}{(x - a)^k}$

• For each irreducible quadratic factor $(x^2 + bx + c)^m$, include terms:

  • $\frac{B_1x + C_1}{x^2 + bx + c} + \frac{B_2x + C_2}{(x^2 + bx + c)^2} + \cdots + \frac{B_mx + C_m}{(x^2 + bx + c)^m}$

Examples of Decomposition Forms

• For $\frac{1}{(x - 1)(x - 2)}$:

  • $\frac{A}{x - 1} + \frac{B}{x - 2}$

• For $\frac{1}{(x - 1)^2}$:

  • $\frac{A}{x - 1} + \frac{B}{(x - 1)^2}$

• For $\frac{1}{x^2 + 1}$:

  • $\frac{Bx + C}{x^2 + 1}$

Solving for Coefficients

After setting up the decomposition, multiply both sides by the denominator to clear fractions, then equate coefficients of like powers of x to solve for the unknowns.

Example 1: Simple Linear Factors

Decompose $\frac{5}{x^2 - 2x - 3}$

• Factor denominator: $x^2 - 2x - 3 = (x - 3)(x + 1)$

• Set up: $\frac{5}{(x - 3)(x + 1)} = \frac{A}{x - 3} + \frac{B}{x + 1}$

• Multiply both sides by $(x - 3)(x + 1)$:

$5 = A(x + 1) + B(x - 3)$

• Expand and solve for A and B by plugging in suitable x-values or equating coefficients.

Example 2: Repeated Linear Factors

Decompose $\frac{4}{(x - 1)^2}$

• Set up: $\frac{4}{(x - 1)^2} = \frac{A}{x - 1} + \frac{B}{(x - 1)^2}$

• Multiply both sides by $(x - 1)^2$:

$4 = A(x - 1) + B$

• Solve for A and B.

Example 3: Irreducible Quadratic Factor

Decompose $\frac{4x}{x^2 + 5x}$

• Factor denominator: $x^2 + 5x = x(x + 5)$

• Set up: $\frac{4x}{x(x + 5)} = \frac{A}{x} + \frac{B}{x + 5}$

• Multiply both sides by $x(x + 5)$:

$4x = A(x + 5) + Bx$

• Solve for A and B.

Example 4: Quadratic Factor

Decompose $\frac{3x^2}{(x - 1)^2(x^2 + 5)}$

• Set up: $\frac{3x^2}{(x - 1)^2(x^2 + 5)} = \frac{A}{x - 1} + \frac{B}{(x - 1)^2} + \frac{Cx + D}{x^2 + 5}$

• Multiply both sides by $(x - 1)^2(x^2 + 5)$ and solve for A, B, C, D.

Summary Table: Partial Fraction Decomposition Forms

Denominator Factor	Partial Fraction Form	Example
Linear, distinct: $(x - a)(x - b)$	$\frac{A}{x - a} + \frac{B}{x - b}$	$\frac{1}{(x - 1)(x - 2)}$
Linear, repeated: $(x - a)^k$	$\frac{A_1}{x - a} + \frac{A_2}{(x - a)^2} + \cdots + \frac{A_k}{(x - a)^k}$	$\frac{1}{(x - 1)^2}$
Irreducible quadratic: $(x^2 + bx + c)$	$\frac{Bx + C}{x^2 + bx + c}$	$\frac{1}{x^2 + 1}$
Repeated quadratic: $(x^2 + bx + c)^m$	$\frac{B_1x + C_1}{x^2 + bx + c} + \cdots + \frac{B_mx + C_m}{(x^2 + bx + c)^m}$	$\frac{1}{(x^2 + 1)^2}$

Applications

• Partial fractions are especially useful for integrating rational functions in calculus.

• They also appear in solving differential equations and in Laplace transforms.

Additional info: The above notes expand on the fragmented original content, providing a complete, self-contained guide to partial fraction decomposition as required in Precalculus.