Problem 33 - Algebra and Trigonometry

Question

\(13-44=\) Find the partial fraction decomposition of the rational function. $$ \frac{-10 x^{2}+27 x-14}{(x-1)^{3}(x+2)} $$

Step-by-Step Solution

Answer

The partial fraction decomposition is \( \frac{3}{x-1} + \frac{-9}{(x-1)^2} + \frac{4}{(x-1)^3} + \frac{6}{x+2} \).

Step 1: Identify the Partial Fraction Form

Since the denominator is \((x-1)^3(x+2)\), we write the partial fraction decomposition form as follows:\[\frac{-10x^2 + 27x - 14}{(x-1)^3(x+2)} = \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{(x-1)^3} + \frac{D}{x+2}\]where \(A\), \(B\), \(C\), and \(D\) are constants to be determined.

Step 2: Clear the Denominator

Multiply both sides of the equation by the common denominator \((x-1)^3(x+2)\) to eliminate fractions:\[-10x^2 + 27x - 14 = A(x-1)^2(x+2) + B(x-1)(x+2) + C(x+2) + D(x-1)^3\]

Step 3: Expand and Simplify

Expand each term on the right-hand side:\(A(x-1)^2(x+2) = A(x^2 - 2x + 1)(x+2)\),\(B(x-1)(x+2) = B(x^2 + x - 2)\),\(C(x+2) = C(x+2)\), and\(D(x-1)^3 = D(x^3 - 3x^2 + 3x - 1)\).Combine these to match the left-hand side polynomial.

Step 4: Collect Like Terms

Collect the like terms of \(x\) on both sides to form a polynomial:\[-10x^2 + 27x - 14 = (A + B + D)x^3 + (2A - 3D)x^2 + (-2A + B + 3C)x + (A - 2B + 2C - D)\]

Step 5: System of Equations

Equate the coefficients of corresponding powers of \(x\) to form a system of equations:- For \(x^3\): \(A + B + D = 0\)- For \(x^2\): \(2A - 3D = -10\)- For \(x\): \(-2A + B + 3C = 27\)- For constant: \(A - 2B + 2C - D = -14\)

Step 6: Solve the System of Equations

Solve the system of equations obtained in the previous step to find the values of \(A\), \(B\), \(C\), and \(D\). - Solving them, we get: - \(A = 3\) - \(B = -9\) - \(C = 4\) - \(D = 6\)

Step 7: Write the Partial Fraction Decomposition

Substitute the values of \(A\), \(B\), \(C\), and \(D\) back into the partial fraction decomposition:\[\frac{-10x^2 + 27x - 14}{(x-1)^3(x+2)} = \frac{3}{x-1} + \frac{-9}{(x-1)^2} + \frac{4}{(x-1)^3} + \frac{6}{x+2}\]

Key Concepts

Rational Function

A rational function is a fraction where both the numerator and the denominator are polynomials. In the form of \(\frac{P(x)}{Q(x)}\), the polynomial \(P(x)\) is known as the numerator, and \(Q(x)\) is the denominator. These functions are very common in algebra and calculus, as they can describe relationships involving rates or birth-to-death processes in population models.
In our exercise, we are working with the rational function: \[\frac{-10x^2 + 27x - 14}{(x-1)^3(x+2)}\]
This is a complicated rational function because both the numerator and denominator have several terms. The degree of the numerator is 2, represented by the highest power of \(x\), and the degree of the denominator is 4, thanks to the term \((x-1)^3(x+2)\). When the degree of the numerator is less than the degree of the denominator, as in this case, it suggests the function is proper, and is a good candidate for partial fraction decomposition.
Rational functions are pivotal for simplifying calculus problems and for solving differential equations where simpler fractions are needed to understand the behavior of the function in detailed analysis.

System of Equations

In algebra, a system of equations is a collection of two or more equations with the same set of variables. Solving a system means finding the values for the variables that satisfy all the given equations simultaneously. These systems are key tools in a lot of mathematical problem solving.
In the given problem, we form a system of equations based on the coefficients of corresponding powers of \(x\) on both sides of the decomposition equality. This comes after we equate power-by-power equivalent terms from both sides of the equation:

• \(x^3: A + B + D = 0\)
• \(x^2: 2A - 3D = -10\)
• \(x: -2A + B + 3C = 27\)
• \(constant: A - 2B + 2C - D = -14\)

The solutions to this system, \(A = 3\), \(B = -9\), \(C = 4\), and \(D = 6\), represent the constants used in the partial fraction decomposition. Solving such systems often involves substitution, elimination, or matrix methods, depending on the complexity and size of the system. Understanding how to effectively set up and solve systems of equations is crucial for breaking down complex algebraic problems into solvable parts.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator are polynomials. They can be simplified, multiplied, divided, or decomposed into simpler fractions, which is a common encounter in algebraic manipulations.
In our case, the challenge was to decompose the algebraic fraction into simpler parts, which helps in simplifying the calculation and integration processes. The fraction

• \(\frac{-10x^2 + 27x - 14}{(x-1)^3(x+2)}\)

can be split into simpler algebraic fractions each having a single polynomial root in the denominator:
\[\frac{3}{x-1} + \frac{-9}{(x-1)^2} + \frac{4}{(x-1)^3} + \frac{6}{x+2}\]
Decomposing a complex algebraic fraction into partial fractions involves writing it as a sum of single terms, each with simpler denominators. This process helps further analyze or integrate them individually, especially in calculus problems. This decomposition is essential in breaking down polynomial relationships and simplifying expressions for easier analysis and execution of operations.