Problem 35 - Algebra and Trigonometry

Question

\(13-44=\) 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

Decompose into \(\frac{A}{x+2} + \frac{B}{(x+2)^2} + \frac{C}{x+3} + \frac{D}{(x+3)^2}\), solve for constants.

Step 1: Identify the Form of the Decomposition

Since the denominator is \( (x+2)^2(x+3)^2 \), it has repeated linear factors. Therefore, the partial fraction decomposition will have the form:\[\frac{A}{x+2} + \frac{B}{(x+2)^2} + \frac{C}{x+3} + \frac{D}{(x+3)^2}\]where \( A, B, C, \) and \( D \) are constants to be determined.

Step 2: Set Up the Partial Fraction Equation

Start by writing the equation for the original rational function equated to the right-hand side of the fractional decomposition:\[\frac{3x^3 + 22x^2 + 53x + 41}{(x+2)^2(x+3)^2} = \frac{A}{x+2} + \frac{B}{(x+2)^2} + \frac{C}{x+3} + \frac{D}{(x+3)^2}\].Multiply through by the denominator \((x+2)^2(x+3)^2\) to eliminate the denominators:\[3x^3 + 22x^2 + 53x + 41 = A(x+2)(x+3)^2 + B(x+3)^2 + C(x+2)^2(x+3) + D(x+2)^2.\]

Step 3: Expand Each Term

Expand each term on the right-hand side to obtain a single polynomial expression.- Expand \( A(x+2)(x+3)^2 \):\[A((x+3)^2(x+2)) = A(x^3 + 6x^2 + 11x + 6)\]- Expand \( B(x+3)^2 \):\[B(x^2 + 6x + 9)\]- Expand \( C(x+2)^2(x+3) \):\[C(x^3 + 4x^2 + 4x)\]- Expand \( D(x+2)^2 \):\[D(x^2 + 4x + 4)\].Combine all the terms to formulate one polynomial expression.

Step 4: Combine and Equate Coefficients

The full expansion becomes:\[Ax^3 + 6Ax^2 + 11Ax + 6A + Bx^2 + 6Bx + 9B + Cx^3 + 4Cx^2 + 4Cx + Dx^2 + 4Dx + 4D.\]Combine like terms:\[(A + C)x^3 + (6A + B + 4C + D)x^2 + (11A + 6B + 4C + 4D)x + (6A + 9B + 4D).\]Equate the coefficients of like powers of \(x\) from both sides with the left-hand side polynomial \(3x^3 + 22x^2 + 53x + 41\).

Step 5: Solve the System of Equations

You obtain the system of equations:- \( A + C = 3 \)- \( 6A + B + 4C + D = 22 \)- \( 11A + 6B + 4C + 4D = 53 \)- \( 6A + 9B + 4D = 41 \).Solve these simultaneous equations to find the values of \( A, B, C, \) and \( D \).

Step 6: Write the Final Partial Fraction Decomposition

After solving the equations, substitute the values of \( A, B, C, \) and \( D \) back into the decomposition form:\[\frac{A}{x+2} + \frac{B}{(x+2)^2} + \frac{C}{x+3} + \frac{D}{(x+3)^2}.\]This gives the final partial fraction decomposition of the rational function.

Key Concepts

Rational Functions

Rational functions are expressions formed by dividing two polynomials. These functions look like fractions, where the numerator and the denominator are polynomial expressions. In the given exercise, the rational function is \( \frac{3x^3 + 22x^2 + 53x + 41}{(x+2)^2(x+3)^2} \). Here, the numerator \(3x^3 + 22x^2 + 53x + 41\) is a polynomial of degree 3, and the denominator \((x+2)^2(x+3)^2\) is also a polynomial, constructed from two factors raised to the power of 2 each.
The goal of partial fraction decomposition is to express this complex expression as a sum of simpler, more manageable fractions, each with a single factor in the denominator, making it easier to integrate or work with further.

Repeated Linear Factors

When the denominator of a rational function consists of factors raised to a power greater than one, these are called repeated linear factors. In our case \((x+2)^2(x+3)^2\), both \(x+2\) and \(x+3\) are repeated linear factors since they appear squared.
To decompose a rational function with repeated linear factors, we need to assign a separate partial fraction to each power of the factor. This is why the decomposition form in the solution has terms like \(\frac{A}{x+2}\) and \(\frac{B}{(x+2)^2}\). Similarly, \(\frac{C}{x+3}\) and \(\frac{D}{(x+3)^2}\). This allows us to break down the complexity of the rational function into simpler terms that sum up to the original expression.

System of Equations

To find the constants in the partial fraction decomposition, we set up a system of equations. This involves equating the expanded form of the decomposition to the original polynomial and matching coefficients of corresponding powers of \(x\). In this exercise, after expanding and collecting like terms, we get:

• \( A + C = 3 \)
• \( 6A + B + 4C + D = 22 \)
• \( 11A + 6B + 4C + 4D = 53 \)
• \( 6A + 9B + 4D = 41 \)

Solving these equations simultaneously will provide the values of \(A\), \(B\), \(C\), and \(D\). Each solution provides the coefficients needed to complete the partial fraction decomposition.

Polynomial Long Division

Polynomial long division is a method sometimes needed before decomposition when the degree of the numerator is greater than or equal to the degree of the denominator. This step simplifies the polynomial to ensure the numerator is of a lower degree, which is essential for partial fraction decomposition.
However, in this particular exercise, the numerator \(3x^3 + 22x^2 + 53x + 41\) is already of lower degree than the denominator \((x+2)^2(x+3)^2\). Therefore, polynomial long division is not necessary here, as the condition is naturally met. This allows us to directly proceed with expressing the rational function as a sum of partial fractions without altering the given expression.