Problem 28 - Precalculus Mathematics for Calculus

Question

Find the partial fraction decomposition of the rational function. $$\frac{3 x^{2}+5 x-13}{(3 x+2)\left(x^{2}-4 x+4\right)}$$

Step-by-Step Solution

Answer

\(\frac{1}{3x + 2} + \frac{-3}{(x-2)^2}\).

Step 1: Identify the Denominator

The denominator of the given rational function is \((3x + 2)(x^2 - 4x + 4)\). The factors are \(3x + 2\) and \(x^2 - 4x + 4\). Notice \(x^2 - 4x + 4\) can be factored further to \((x-2)^2\). So, the complete factorization of the denominator is \((3x + 2)((x - 2)^2)\).

Step 2: Set Up Partial Fraction

For the rational function \(\frac{3x^2 + 5x - 13}{(3x + 2)((x - 2)^2)}\), we express it as a sum of partial fractions: \[\frac{A}{3x + 2} + \frac{B}{x-2} + \frac{C}{(x-2)^2}\] where \(A\), \(B\), and \(C\) are constants to be determined.

Step 3: Clear the Denominator

Multiply both sides by the denominator \((3x+2)((x-2)^2)\) to eliminate the fractions: \[3x^2 + 5x - 13 = A(x-2)^2 + B(3x+2)(x-2) + C(3x+2)\].

Step 4: Expand and Collect Terms

Expand the right-hand side: \[A(x-2)^2 = A(x^2 - 4x + 4)\]\[B(3x+2)(x-2) = B(3x^2 - 6x + 2x - 4) = B(3x^2 - 4x - 4)\]\[C(3x+2) = 3Cx + 2C\].Combine all terms together on the right-hand side.

Step 5: Equate Coefficients

After combining, align coefficients of like powers of \(x\) on both sides to find A, B, and C:For \(x^2\): \[A + 3B = 3\]For \(x\): \[-4A - 4B + 3C = 5\]For the constant term:\[4A - 4B + 2C = -13\].

Step 6: Solve the System of Equations

Solve the system of equations:1. \(A + 3B = 3\)2. \(-4A - 4B + 3C = 5\)3. \(4A - 4B + 2C = -13\).After solving, \(A = 1\), \(B = 0\), and \(C = -3\).

Step 7: Write the Final Decomposition

Substitute \(A\), \(B\), and \(C\) into the partial fractions: \[\frac{3x^2 + 5x - 13}{(3x + 2)(x-2)^2} = \frac{1}{3x + 2} + \frac{-3}{(x-2)^2}\].This is the partial fraction decomposition of the given rational function.

Key Concepts

Rational Functions

Rational functions are expressions made up of two polynomials. Specifically, they are in the form of a fraction where the numerator and the denominator are both polynomials.
This particular mathematical concept is essential because it allows us to understand a wide array of phenomena, from engineering to natural sciences. An important feature of rational functions is their asymptotes, which occur when the denominator equals zero. These asymptotes can be horizontal, vertical, or even slanted, and they help us grasp the behavior of the function.
For partial fraction decomposition specifically, having rational functions means that we can express these as the sum of simpler fractions. This makes it easier to integrate or manipulate them in calculus and other applications.

Solving Systems of Equations

When performing partial fraction decomposition, we often end up with a collection of equations that need to be solved to find the unknown constants.
This requires solving systems of equations, which is a method used to find the values that satisfy multiple equations simultaneously. These systems can be solved using various methods, including substitution, elimination, or using matrix techniques such as Gaussian elimination.
In this context, the equations come from equating the coefficients of terms with similar powers of the variable. By balancing the equation on both sides, you ensure that each term has the correct coefficient, which can only happen if the unknown constants are accurate. As seen in the given solution, solving the system of equations yielded the values of A, B, and C, which were then used to form the partial fraction expression.

Polynomial Factorization

Polynomial factorization is a crucial step when dealing with rational functions, especially for partial fraction decomposition.
This involves breaking down a complex polynomial into its simpler factors, making it easier to analyze and manipulate. For example, in the given exercise, the polynomial in the denominator was factored into \( (3x+2)((x-2)^2) \). Recognizing \( (x^2 - 4x + 4) \) as \( (x-2)^2 \) is a key step that simplifies the problem.
The process of factorization also reveals roots or zeros of the polynomial, which are useful in graph analysis. It is foundational in calculus, as it allows complex expressions to be integrated or differentiated by simplifying them into more manageable pieces. In practice, familiarize yourself with common factorization techniques such as factoring out the greatest common factor, grouping, and using special formulas like the difference of squares or the sum and difference of cubes.