Problem 32 - Precalculus: Mathematics for Calculus

Question

Find the partial fraction decomposition of the rational function. $$\frac{x^{3}-2 x^{2}-4 x+3}{x^{4}}$$

Step-by-Step Solution

Answer

The decomposition is \( \frac{1}{x} - \frac{2}{x^2} - \frac{4}{x^3} + \frac{3}{x^4} \).

Step 1: Analyze the Denominator

The denominator of the given rational function is \( x^4 \). This is a polynomial raised to the fourth power and it is already factored as much as possible, since it involves no constant factors or additional polynomials.

Step 2: Setup the Partial Fraction

Since the denominator is \( x^4 \), which is a power of \( x \), the partial fraction decomposition will involve terms for every power from 1 to 4. The decomposition will take the form: \[ \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4}. \]

Step 3: Multiply Through by the Denominator

Multiply the entire equation by \( x^4 \) to eliminate the denominators:\[ x^3 - 2x^2 - 4x + 3 = A x^3 + B x^2 + C x + D. \]

Step 4: Equate Coefficients

Compare the coefficients of corresponding powers of \( x \) on both sides of the equation:- For \( x^3 \), equate: \( 1 = A \)- For \( x^2 \), equate: \( -2 = B \)- For \( x \), equate: \( -4 = C \)- For the constant term, equate: \( 3 = D \)

Step 5: Substitute and Solve for Constants

From equating coefficients, we have direct values for \( A, B, C, \) and \( D \):- \( A = 1 \)- \( B = -2 \)- \( C = -4 \)- \( D = 3 \)

Step 6: Write Final Decomposition

Substitute these values back into the partial fraction expression:\[ \frac{x^3 - 2x^2 - 4x + 3}{x^4} = \frac{1}{x} + \frac{-2}{x^2} + \frac{-4}{x^3} + \frac{3}{x^4}. \]

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials. In mathematics, these functions are incredibly versatile and appear frequently in various fields. In this specific case, the rational function is \( \frac{x^3-2x^2-4x+3}{x^4} \). Here, the numerator \( x^3-2x^2-4x+3 \) is a third-degree polynomial, while the denominator \( x^4 \) is a fourth-degree polynomial.
This means our rational function is proper because the degree of the numerator is less than the degree of the denominator. Proper rational functions often need techniques like partial fraction decomposition to break them into simpler components, making them easier to integrate or simplify further. Understanding rational functions gives us a powerful tool to untangle seemingly complex algebraic expressions.
In this context, partial fraction decomposition helps us express the function as a sum of simpler fractions, aiding in mathematical operations like integration.

Coefficients Comparison

The technique of equating coefficients is crucial when decomposing rational functions. Once we set up the partial fraction such as \( \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4} \), we need to multiply through by the common denominator to clear out the fractions. This action converts our equation to \( x^3 - 2x^2 - 4x + 3 = Ax^3 + Bx^2 + Cx + D \). By doing this, we can easily see that each side must essentially be the same polynomial.
As a result, each coefficient (the number before each power of \( x \)) on the left must equal each corresponding coefficient on the right. This comparison forms a system of equations:

• \( 1 = A \)
• \(-2 = B \)
• \(-4 = C \)
• \(3 = D \)

Solving this set of equations by inspection gives us the values for \( A, B, C, \) and \( D \). This step-by-step approach of equating coefficients is highly reliable and systematic, making it ideal for students tackling partial fractions.

Denominator Analysis

Analyzing the denominator is the first step in partial fraction decomposition. The denominator tells us how to set up our partial fractions. For our function, the simple power of \( x \) raised to the fourth degree, \( x^4 \), indicates that we should include fractions for all these powers.
Understanding the denominator helps us determine the number of fractions and their forms. In the equation \( \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4} \), each fraction represents a term for each power of \( x \) from one to four. Analyzing the denominator's simplest form guides the decomposition process and ensures comprehensiveness.
This thoughtful setup is particularly important when dealing with more complex denominators that consist of multiple factors. For any partial fraction decomposition, proper analysis of the denominator steers us in the right direction, laying a solid foundation for equating coefficients and solving for unknowns.