Problem 33 - Precalculus: Mathematical for Calculus

Question

Find the partial fraction decomposition of the rational function. $$\frac{3 x^{3}+22 x^{2}+53 x+41}{(x+2)^{2}(x+3)^{2}}$$

Step-by-Step Solution

Answer

The partial fraction decomposition is \(\frac{5}{x+2} + \frac{-4}{(x+2)^2} + \frac{1}{x+3} + \frac{10}{(x+3)^2}\).

Step 1: Set up the Partial Fraction Form

To begin the partial fraction decomposition, represent the given rational function as a sum of fractions with unknown coefficients over the factors of the denominator. Since the denominator \((x+2)^2(x+3)^2\), the partial fraction form is as follows:\[\frac{3x^3 + 22x^2 + 53x + 41}{(x+2)^2(x+3)^2} = \frac{A}{x+2} + \frac{B}{(x+2)^2} + \frac{C}{x+3} + \frac{D}{(x+3)^2}\]Here, \(A\), \(B\), \(C\), and \(D\) are constants to be determined.

Step 2: Clear the Denominator

Multiply both sides by the common denominator \((x+2)^2(x+3)^2\) to eliminate fractions:\[3x^3 + 22x^2 + 53x + 41 = A(x+2)(x+3)^2 + B(x+3)^2 + C(x+3)(x+2)^2 + D(x+2)^2\]

Step 3: Expand the Right Side

Expand the expressions on the right-hand side to simplify and match the coefficients with the left-hand side:1. \(A(x+2)(x+3)^2 = A(x+9x^2+27x+18) = A(x^3+9x^2+27x+18x)\)2. \(B(x+3)^2 = B(x^2+6x+9)\)3. \(C(x+3)(x+2)^2 = C(x+4x^2+8x+4)\)4. \(D(x+2)^2 = D(x^2+4x+4)\)

Step 4: Collect and Equate Coefficients

Expand the products and combine all the terms on the right-hand side of the equation. Then collect like terms and equate them to the corresponding coefficients from the left side \(3x^3 + 22x^2 + 53x + 41\). This will produce a system of equations for \(A\), \(B\), \(C\), \(D\).

Step 5: Solve the System of Equations

Solve the system of equations obtained from the coefficient comparison:- Coefficient of \(x^3\): \( A + C = 3 \)- Coefficient of \(x^2\): \(9A + 12C + B + D = 22\)- Coefficient of \(x\): \(27A + 8C + 6B + 4D = 53\)- Constant term: \(18A + 4C + 9B + 4D = 41\)By solving this system, you will find the values of \(A\), \(B\), \(C\), and \(D\).

Step 6: Write the Final Decomposition

After solving the system, substitute the values of \(A\), \(B\), \(C\), and \(D\) back into the partial fraction form:\[\frac{3x^3 + 22x^2 + 53x + 41}{(x+2)^2(x+3)^2} = \frac{A}{x+2} + \frac{B}{(x+2)^2} + \frac{C}{x+3} + \frac{D}{(x+3)^2}\]

Key Concepts

Rational Functions

Rational functions are a fundamental concept in algebra. They are ratios of two polynomial expressions, where the numerator and the denominator are both polynomials. A typical example of a rational function is \( \frac{3x^3 + 22x^2 + 53x + 41}{(x+2)^2(x+3)^2} \).Rational functions can exhibit a variety of behaviors, such as asymptotes, intercepts, and discontinuities, depending on the degree of the numerator and the denominator. Understanding these functions is crucial as they appear frequently in calculus and real-world problems.
When dealing with rational functions in problems like partial fraction decomposition, we're particularly interested in breaking them down into simpler fractions. These simpler fractions can often be integrated or differentiated more easily. The goal is to express a complex rational function as a sum of simpler fractions with linear or quadratic denominators.

Denominators

The denominator of a rational function is the polynomial located beneath the numerator in a fraction. For the given function \( \frac{3x^3 + 22x^2 + 53x + 41}{(x+2)^2(x+3)^2} \), the denominator is \( (x+2)^2(x+3)^2 \).Denominators significantly influence the behavior of rational functions because they determine where the function is undefined (where they equal zero). These are the poles or vertical asymptotes of the function.
In partial fraction decomposition, knowing the structure of the denominator is crucial. You must express the rational function as a series of simpler fractions whose denominators consist of the factors of the original rational function's denominator. This step prepares the function for decomposition into partial fractions, which can then be manipulated individually.

System of Equations

A system of equations is used to find the unknown coefficients in the partial fraction decomposition of a rational function. For such decompositions, the equation with multiple terms on both sides is simplified through comparison of coefficients.Consider the expression from the problem: \( 3x^3 + 22x^2 + 53x + 41 = A(x+2)(x+3)^2 + B(x+3)^2 + C(x+3)(x+2)^2 + D(x+2)^2 \). By expanding and comparing coefficients of like terms (such as \( x^3, x^2, x \) and the constant term), you'll arrive at a set of linear equations:

• \( A + C = 3 \)
• \( 9A + 12C + B + D = 22 \)
• \( 27A + 8C + 6B + 4D = 53 \)
• \( 18A + 4C + 9B + 4D = 41 \)

Solving these equations simultaneously allows you to find the precise values of \( A, B, C, \) and \( D \), thereby giving you the exact partial fraction decomposition. This combination of algebraic techniques helps to understand and solve complex expressions by breaking them into manageable parts.