How to Compare Coefficients in Partial Fractions
Partial fractions are a useful technique in algebra and calculus for simplifying complex rational expressions. When dealing with rational functions, it is often necessary to express them as a sum of simpler fractions. This process, known as partial fraction decomposition, involves comparing coefficients to ensure that the original and decomposed expressions are equivalent. In this article, we will discuss how to compare coefficients in partial fractions and provide some examples to illustrate the process.
Understanding the Basics
Before diving into the comparison of coefficients, it is essential to understand the basics of partial fractions. A rational expression is a fraction where both the numerator and denominator are polynomials. When a rational expression cannot be simplified further, it may be expressed as a sum of partial fractions. The general form of a partial fraction decomposition is:
\[ \frac{P(x)}{Q(x)} = \frac{A_1}{x – a_1} + \frac{A_2}{x – a_2} + \ldots + \frac{A_n}{x – a_n} \]
where \( P(x) \) is the numerator, \( Q(x) \) is the denominator, and \( A_1, A_2, \ldots, A_n \) are constants representing the coefficients.
Comparing Coefficients
To compare coefficients in partial fractions, follow these steps:
1. Multiply both sides of the equation by the denominator, \( Q(x) \), to eliminate the fractions.
2. Expand the resulting expression to obtain a polynomial equation.
3. Equate the coefficients of the corresponding powers of \( x \) on both sides of the equation.
4. Solve the resulting system of equations to find the values of the coefficients \( A_1, A_2, \ldots, A_n \).
Example 1
Consider the following rational expression:
\[ \frac{2x^2 + 5x + 2}{x^2 – 1} \]
To decompose this expression into partial fractions, we first multiply both sides by the denominator:
\[ 2x^2 + 5x + 2 = (A_1 + A_2x)(x + 1)(x – 1) \]
Expanding the right-hand side, we get:
\[ 2x^2 + 5x + 2 = A_1(x^2 – 1) + A_2x(x^2 – 1) \]
Now, we equate the coefficients of the corresponding powers of \( x \):
\[ A_1 = 2 \]
\[ A_2 = 5 \]
Therefore, the partial fraction decomposition of the given expression is:
\[ \frac{2x^2 + 5x + 2}{x^2 – 1} = \frac{2}{x + 1} + \frac{5}{x – 1} \]
Example 2
Consider the following rational expression:
\[ \frac{3x^3 – 2x^2 + 5x – 1}{x^2 + 2x + 1} \]
To decompose this expression into partial fractions, we first multiply both sides by the denominator:
\[ 3x^3 – 2x^2 + 5x – 1 = (A_1 + A_2x + A_3x^2)(x^2 + 2x + 1) \]
Expanding the right-hand side, we get:
\[ 3x^3 – 2x^2 + 5x – 1 = A_1(x^2 + 2x + 1) + A_2x(x^2 + 2x + 1) + A_3x^2(x^2 + 2x + 1) \]
Now, we equate the coefficients of the corresponding powers of \( x \):
\[ A_1 = -1 \]
\[ A_2 = 3 \]
\[ A_3 = 0 \]
Therefore, the partial fraction decomposition of the given expression is:
\[ \frac{3x^3 – 2x^2 + 5x – 1}{x^2 + 2x + 1} = \frac{-1}{x + 1} + \frac{3}{x} \]
In conclusion, comparing coefficients in partial fractions is a crucial step in the decomposition process. By following the steps outlined in this article, you can successfully decompose rational expressions into simpler fractions and simplify complex problems in algebra and calculus.
