Maths STD12

Partial Fractions

August 10, 2023
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Partial fraction decomposition is a mathematical technique used to break down a complex rational function into a sum of simpler rational expressions. It’s particularly useful when working with integrals in calculus, solving differential equations, and in various applications within engineering and physics.

Here’s a breakdown of what it entails:

– **Rational Function**: A rational function is a fraction where both the numerator and denominator are polynomials. For example, \frac{P(x)}{Q(x)}, where P(x) and Q(x) are polynomials.

– **Proper Rational Function**: Partial fractions are generally applied to proper rational functions, where the degree of the numerator is less than the degree of the denominator.

– **Factorizing the Denominator**: The first step in partial fraction decomposition is to factorize the denominator into linear and/or irreducible quadratic factors.

– **Writing Partial Fractions**: Depending on the factors in the denominator, the rational function is expressed as a sum of fractions with simpler denominators. For example, if the denominator has distinct linear factors, you can write it as \frac{A}{(ax + b)}.

– **Finding Coefficients**: Coefficients in the partial fractions are found by various methods such as equating coefficients or substituting specific values of x.

– **Applications**: This decomposition makes it easier to perform operations such as integration, differentiation, and simplification of complex expressions.

The process of partial fraction decomposition turns an intimidating rational function into a more manageable expression, making various mathematical tasks more tractable. It’s a powerful tool in calculus and higher-level mathematics.

Concepts

– **Definition**: Partial fraction decomposition is a technique to express a rational function as a sum of simpler rational expressions. It is particularly useful in integrating rational functions.

– **Applicability**: This method applies to rational functions, where the degree of the numerator is less than the degree of the denominator.

– **Factorization**: The first step is to factorize the denominator into linear and/or irreducible quadratic factors.

– **Form Identification**: Depending on the factors in the denominator, you can write down the general form of the partial fractions. Here are some forms:
– For distinct linear factors: \frac{A}{(ax + b)}
– For repeated linear factors: \frac{A}{(ax + b)^n}
– For irreducible quadratic factors: \frac{Ax + B}{(ax^2 + bx + c)}

– **Solving for Coefficients**: You can find the coefficients (like A, B, etc.) by clearing the denominators, expanding, and equating coefficients. Sometimes, using specific values of x to create a system of equations can be helpful.

– **Integration Applications**: Partial fraction decomposition is widely used in calculus to simplify integrals of rational functions.

– **Covered Under Complex Analysis**: In more advanced mathematics, this method can be extended to cover complex functions and residues, proving to be a powerful tool in complex analysis.

– **Use in Control Systems and Signal Processing**: It plays a crucial role in systems theory, particularly in solving linear differential equations, Laplace transforms, and analyzing systems’ behavior.

– **Software Tools**: Many computational tools like Mathematica, MATLAB, or SymPy in Python can automate the process of finding partial fraction decompositions.

– **Common Missteps**: Errors in factoring the denominator or misidentifying the correct form for the decomposition can lead to mistakes. Careful attention to these details is essential for the correct application of the method.

Understanding these notes can help in the efficient application of partial fraction decomposition in various mathematical contexts.

Working Rule

1. **Check Degree**: Make sure the degree of the numerator is less than the degree of the denominator. If not, perform polynomial division first.

2. **Factorize the Denominator**: Factor the denominator into irreducible factors, which can be linear or quadratic.

3. **Identify the Form**: Write down the general form of the partial fraction decomposition based on the factors in the denominator:
– For a non-repeated linear factor (ax + b), use \frac{A}{(ax + b)}.
– For a repeated linear factor (ax + b)^n, use \frac{A_1}{(ax + b)} + \frac{A_2}{(ax + b)^2} + \ldots + \frac{A_n}{(ax + b)^n}.
– For an irreducible quadratic factor (ax^2 + bx + c), use \frac{Ax + B}{(ax^2 + bx + c)}.

4. **Clear the Denominators**: Multiply both sides of the equation by the common denominator to clear fractions. This will result in a polynomial equation.

5. **Solve for Coefficients**: By expanding and collecting like terms, you can solve for the unknown coefficients by:
– Equating coefficients for the corresponding powers of x.
– Substituting specific values for x to create a system of linear equations.

6. **Reassemble the Expression**: Combine the solved coefficients with the previously identified forms to reassemble the expression as the sum of partial fractions.

7. **Verify the Solution**: It can be beneficial to verify your solution by combining the partial fractions and comparing it with the original rational function.

8. **Integration (If Applicable)**: If the purpose of decomposition is to integrate the expression, you can now integrate each term individually, which will typically be simpler than integrating the original rational function.

Remember, not every rational function can be decomposed into partial fractions; the method is specifically useful for proper rational functions, where the degree of the numerator is less than the degree of the denominator.

Worksheet

Certainly! Here are 12 multiple-choice questions related to partial fraction decomposition:

1. **Question**: Decompose \frac{4}{x^2 - 1}:
**Options**:
A) \frac{2}{x+1} + \frac{2}{x-1}
B) \frac{2}{x+1} - \frac{2}{x-1}
C) \frac{1}{x+1} + \frac{3}{x-1}
D) \frac{4}{x+1} - \frac{4}{x-1}
**Correct Answer**: A

2. **Question**: What is the decomposition of \frac{3x+2}{x^2-4x+3}?
**Options**:
A) \frac{1}{x-1} + \frac{2}{x-3}
B) \frac{2}{x-1} + \frac{1}{x-3}
C) \frac{1}{x-1} - \frac{2}{x-3}
D) \frac{3}{x-1} + \frac{2}{x-3}
**Correct Answer**: B

3. **Question**: Decompose \frac{5x+3}{(x+2)(x-3)^2}:
**Options**:
A) \frac{1}{x+2} + \frac{2}{x-3} + \frac{1}{(x-3)^2}
B) \frac{2}{x+2} + \frac{1}{x-3} + \frac{2}{(x-3)^2}
C) \frac{1}{x+2} - \frac{1}{x-3} + \frac{3}{(x-3)^2}
D) \frac{2}{x+2} - \frac{1}{x-3} + \frac{1}{(x-3)^2}
**Correct Answer**: A

4. **Question**: For \frac{1}{x^2+3x+2}, what are the correct partial fractions?
**Options**:
A) \frac{1}{x+1} - \frac{1}{x+2}
B) \frac{1}{x+1} + \frac{1}{x+2}
C) \frac{1}{x-1} - \frac{1}{x+2}
D) \frac{1}{x-1} + \frac{1}{x+2}
**Correct Answer**: A

5. **Question**: Decompose \frac{2x^2+3x-2}{x^3+x^2-6x}:
**Options**:
A) \frac{1}{x} + \frac{1}{x+3} - \frac{2}{x-2}
B) \frac{1}{x} - \frac{2}{x+3} + \frac{1}{x-2}
C) \frac{2}{x} + \frac{1}{x+3} - \frac{1}{x-2}
D) \frac{2}{x} - \frac{3}{x+3} + \frac{1}{x-2}
**Correct Answer**: C

6. **Question**: What is the partial fraction of \frac{4x}{(x+2)^2}?
**Options**:
A) \frac{2}{x+2} + \frac{2}{(x+2)^2}
B) \frac{1}{x+2} + \frac{3}{(x+2)^2}
C) \frac{3}{x+2} + \frac{1}{(x+2)^2}
D) \frac{4}{x+2} + \frac{4}{(x+2)^2}
**Correct Answer**: C

7. **Question**: Decompose \frac{x^2+2x+1}{x^2+x-6}:
**Options**:
A) \frac{1}{x-2} + \frac{2}{x+3}
B) \frac{2}{x-2} + \frac{1}{x+3}
C) \frac{1}{x-2} - \frac{1}{x+3}
D) \frac{2}{x-2} - \frac{2}{x+3}
**Correct Answer**: B

8. **Question**: What is the decomposition of \frac{5}{(x-1)(x^2+1)}?
**Options**:
A) \frac{2}{x-1} + \frac{3}{x^2+1}
B) \frac{3}{x-1} - \frac{2}{x^2+1}
C) \frac{3}{x-1} + \frac{2x}{x^2+1}
D) \frac{2}{x-1} - \frac{3x}{x^2+1}
**Correct Answer**: C

9. **Question**: Decompose \frac{4x^2 - 2x + 6}{x^3 - 3x}:
**Options**:
A) \frac{2}{x} + \frac{1}{x+3} - \frac{3}{x-3}
B) \frac{2}{x} - \frac{2}{x+3} + \frac{1}{x-3}
C) \frac{2}{x} + \frac{2}{x+3} - \frac{2}{x-3}
D) \frac{1}{x} + \frac{3}{x+3} - \frac{2}{x-3}
**Correct Answer**: C

10. **Question**: What is the partial fraction of \frac{3x+1}{(x-4)(x+5)}?
**Options**:
A) \frac{1}{x-4} + \frac{3}{x+5}
B) \frac{3}{x-4} + \frac{1}{x+5}
C) \frac{1}{x-4} - \frac{2}{x+5}
D) \(\frac{2}{x-4} – \frac{1}{x+