Partial Fraction Revision Notes BTER Polytechnic Classes

 

Partial Fractions: Detailed Notes

Partial fraction decomposition is an essential technique in algebra, especially in integral calculus, differential equations, and control systems. It simplifies complex rational expressions into simpler fractions, which are easier to work with, particularly when integrating or differentiating.

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4.1 Definition of Polynomial Fraction, Proper & Improper Fractions

A polynomial fraction is a rational expression of the form:

$$\frac{P(x)}{Q(x)}$$

Where $P(x)$ and $Q(x)$ are polynomials.

Proper Fraction:

A proper fraction is a fraction where the degree of the numerator polynomial $P(x)$ is less than the degree of the denominator polynomial $Q(x)$. Mathematically, this means:

$$\text{Degree of } P(x) < \text{Degree of } Q(x)$$

Example:

$$\frac{x+2}{x^2 + 1} \quad (\text{Degree of numerator is 1, and degree of denominator is 2})$$

Improper Fraction:

An improper fraction is a fraction where the degree of the numerator polynomial $P(x)$ is greater than or equal to the degree of the denominator polynomial $Q(x)$. Mathematically, this means:

$$\text{Degree of } P(x) \geq \text{Degree of } Q(x)$$

Example:

$$\frac{x^2 + 1}{x + 1} \quad (\text{Degree of numerator is 2, and degree of denominator is 1})$$

To handle improper fractions, we first perform polynomial division to express the improper fraction as a sum of a polynomial and a proper fraction. The proper fraction can then be decomposed into partial fractions.

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4.2 Definition of Partial Fractions

Partial fractions are a way of breaking down a rational function (fraction of polynomials) into a sum of simpler fractions. The goal is to decompose a rational function into a sum of fractions where the denominators are simpler polynomials, typically linear or quadratic factors.

For example, the fraction $\frac{P(x)}{Q(x)}$ is broken into several simpler fractions that have linear or quadratic denominators, making it easier to integrate or differentiate.

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4.3 To Resolve Proper Fraction into Partial Fraction with Denominator Containing Non-Repeated Linear Factors

When the denominator contains distinct linear factors, the proper fraction can be expressed as a sum of simpler fractions where each fraction has a denominator corresponding to one of the linear factors.

Steps for Decomposition:

1. Factor the denominator: Write the denominator $Q(x)$ as a product of distinct linear factors. For example, if $Q(x) = (x - a)(x - b)$, the denominator is a product of two distinct linear factors.

2. Set up the partial fraction decomposition: For each distinct linear factor $(x - a)$, assign a constant $\frac{A}{x - a}$. Do the same for each distinct factor in the denominator. For example:

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

3. Solve for the constants $A$ and $B$ by multiplying both sides of the equation by the common denominator $(x - a)(x - b)$, then equating the coefficients of like powers of $x$.

Example:

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

Step 1: Set up the decomposition:

$$\frac{3x + 5}{(x - 1)(x + 2)} = \frac{A}{x - 1} + \frac{B}{x + 2}$$

Step 2: Multiply through by $(x - 1)(x + 2)$ to eliminate the denominators:

$$3x + 5 = A(x + 2) + B(x - 1)$$

Step 3: Expand both sides:

$$3x + 5 = A(x) + 2A + B(x) - B$$ $$3x + 5 = (A + B)x + (2A - B)$$

Step 4: Equate the coefficients of like powers of $x$:

• For $x$: $A + B = 3$
• For constant term: $2A - B = 5$

Step 5: Solve the system of equations:

$$A + B = 3$$ $$2A - B = 5$$

Solving this system, we get:

$$A = 2, \quad B = 1$$

Thus, the partial fraction decomposition is:

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

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4.4 To Resolve Proper Fraction into Partial Fraction with Denominator Containing Repeated Linear Factors

When the denominator contains repeated linear factors, we set up a slightly different form for the partial fraction decomposition.

Steps for Decomposition:

1. Factor the denominator: Write the denominator $Q(x)$ as a product of repeated linear factors. For example, if $Q(x) = (x - a)^2(x - b)$, the denominator has a repeated linear factor $(x - a)^2$.

2. Set up the partial fraction decomposition: For each repeated linear factor $(x - a)^n$, assign a constant for each power of $(x - a)$. The general form is:

   $$\frac{P(x)}{(x - a)^n} = \frac{A_1}{x - a} + \frac{A_2}{(x - a)^2} + \cdots + \frac{A_n}{(x - a)^n}$$

   For example, if $Q(x) = (x - 1)^2(x + 2)$, then:

   $$\frac{P(x)}{(x - 1)^2(x + 2)} = \frac{A}{x - 1} + \frac{B}{(x - 1)^2} + \frac{C}{x + 2}$$

3. Solve for the constants $A, B, C, \dots$ by multiplying both sides by the common denominator $(x - 1)^2(x + 2)$ and then equating the coefficients of like powers of $x$.

Example:

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

Step 1: Set up the decomposition:

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

Step 2: Multiply through by $(x - 1)^2(x + 2)$:

$$x + 1 = A(x - 1)(x + 2) + B(x + 2) + C(x - 1)^2$$

Step 3: Expand both sides:

$$x + 1 = A(x^2 + x - 2) + B(x + 2) + C(x^2 - 2x + 1)$$ $$x + 1 = A(x^2 + x - 2) + B(x + 2) + C(x^2 - 2x + 1)$$

Step 4: Combine like terms, and solve for $A$, $B$, and $C$.

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Summary of Key Points:

1. Proper Fraction: Degree of numerator < Degree of denominator.
2. Improper Fraction: Degree of numerator ≥ Degree of denominator.
3. Partial Fraction Decomposition:
   • If the denominator has distinct linear factors: $\frac{A}{x - a} + \frac{B}{x - b}$
   • If the denominator has repeated linear factors: $\frac{A}{x - a} + \frac{B}{(x - a)^2} + \cdots$
4. Steps:
   • Factor the denominator.
   • Set up partial fractions with undetermined constants.
   • Multiply by the common denominator and equate the coefficients of like powers of $x$.
   • Solve for the constants.

These methods of decomposition are essential in solving complex rational expressions, particularly when integrating or simplifying expressions in calculus and differential equations.