Expressing Rational Functions in Partial Fractions: A Detailed Guide
In this guide, we will explore the process of decomposing a rational function into partial fractions. Specifically, we will tackle the expression 3x2x1 / x x12 using the method of partial fraction decomposition. This method is widely used in calculus and other advanced mathematical applications, and understanding it is crucial for solving a variety of complex algebraic equations.
Introduction to Partial Fraction Decomposition
Partial fraction decomposition is the process of expressing one rational function as the sum of simpler rational functions. This technique is particularly useful when dealing with complex algebraic expressions, as it simplifies the integrals and other operations that may be applied to them. To begin, let’s understand the given expression and its components:
The expression we are working with is: 3x2x1 / x x12. Here, we have a rational function with a quadratic term in the numerator and a linear term in the denominator.
Step-by-Step Process
Step 1: Simplify the Expression
First, let’s simplify the given expression. Notice that x2x1 can be written as x3 and x x12 can be written as x3. Therefore, the expression simplifies to:
3x3 / x3
This simplifies further to:
3
Step 2: Analyze the Simplified Expression
After simplification, the expression becomes a constant. However, for the sake of explanation, let’s consider a general rational function and the steps we would follow:
Let’s assume the expression is of the form A / x2 B / x C / (x 1)2. Here, the goal is to find the values of A, B, and C by equating the simplified numerator to the original expression.
The Process of Equating Coefficients
Let's revisit the original expression 3x2x1 / x x12. We will now proceed with the general method of partial fraction decomposition. The expression can be decomposed as follows:
A / x2 B / x C / (x 1)2
Multiplying both sides by the common denominator, which is x2(x 1)2, we get:
A(x 1)2 Bx(x 1) Cx2 3x2x1
This simplifies to:
A(x2 2x 1) Bx(x 1) Cx2 3x3
Now, expand and simplify the left side:
A(x2 2x 1) Bx2 Bx Cx2 3x3
Ax2 2Ax A Bx2 Bx Cx2 3x3
3x3 (A B C)x2 (2A B)x A 3 2 0
Now, equate the coefficients on both sides of the equation:
3 0
A B C 0
2A B 0
A 0
From the above equations, we can see that A 0, and substituting A 0, we get B 0, and thus C 0. Therefore, the expression reduces to:
3 / (x2) 0 0
Conclusion
In conclusion, the process of partial fraction decomposition involves simplifying the given rational function, identifying the form of the partial fractions, and then equating the coefficients to find the constants. In this specific example, the expression 3x2x1 / x x12 simplifies to 3, and the decomposition process confirms this simplification.
This method is invaluable for solving intricate algebraic problems, and understanding it thoroughly enhances the problem-solving skills in mathematics and calculus. Whether you are a student or a professional in the field of mathematics, mastering partial fraction decomposition is a crucial step towards advanced problem-solving skills.