Problem 36 - Algebra and Trigonometry

Question

Find the partial fraction decomposition of the rational function. \(\frac{3 x^{2}-2 x+8}{x^{3}-x^{2}+2 x-2}\)

Step-by-Step Solution

Answer

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

Step 1: Factor the Denominator

First, let's factor the denominator \(x^{3}-x^{2}+2x-2\). By grouping, we get \((x^3-x^2) + (2x-2)\). Factor each group to get \(x^2(x-1) + 2(x-1)\). Notice \(x-1\) is a common factor, so we can factor it out: \((x-1)(x^2+2)\).

Step 2: Set up the Partial Fractions

With the factored denominator \((x-1)(x^2+2)\), the partial fraction decomposition will be of the form: \(\frac{A}{x-1} + \frac{Bx + C}{x^2 + 2}\).

Step 3: Clear the Denominator

Multiply both sides by the original denominator \((x-1)(x^2+2)\) to clear the fractions: \[3x^2 - 2x + 8 = A(x^2+2) + (Bx + C)(x-1)\].

Step 4: Expand and Collect Like Terms

Expand the right side: \(A(x^2+2) = Ax^2 + 2A\), \((Bx + C)(x-1) = Bx^2 + (C-B)x - C\). Combine: \[3x^2 - 2x + 8 = (A + B)x^2 + (C-B)x + (2A-C)\].

Step 5: Equate Coefficients

Equate the coefficients of the same powers of \(x\) from both sides: \(A + B = 3\), \(C - B = -2\), \(2A - C = 8\).

Step 6: Solve the System of Equations

Solve these equations: From \(A + B = 3\) and \(C - B = -2\): 1. \(C = B - 2\)Substitute \(C = B - 2\) into \(2A - C = 8\), gives:2. \(2A - (B - 2) = 8\) ⟹ \(2A - B + 2 = 8\)Therefore, \(2A - B = 6\).Now solve the system:a. \(A + B = 3\)b. \(2A - B = 6\)From a, \(B = 3 - A\). Substitute into b: \(2A - (3 - A) = 6\).Solve for \(A\):\[3A - 3 = 6\]\[3A = 9\]\[A = 3\]Substitute \(A = 3\) into \(B = 3 - A\):\[B = 3 - 3 = 0\]Finally, substitute \(B = 0\) into \(C = B - 2\):\[C = 0 - 2 = -2\].

Step 7: Write the Partial Fraction Decomposition

With \(A = 3\), \(B = 0\), and \(C = -2\), the partial fraction decomposition of the rational function is: \[ \frac{3x^2 - 2x + 8}{x^3 - x^2 + 2x - 2} = \frac{3}{x-1} - \frac{2}{x^2+2} \].

Key Concepts

Rational Functions

Rational functions are expressions defined as the quotient of two polynomials. The form of a rational function is typically written as \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). These functions can exhibit complex behavior such as asymptotes and discontinuities, depending on the roots of the denominator.

Understanding rational functions is essential for many areas of mathematics, including calculus and algebra, as they often model real-world scenarios where a variable's rate of change is dependent on another variable.

The primary step in solving problems involving rational functions, like partial fraction decomposition, is to ensure the denominator is non-zero. Also, evaluating their limits and behaviors can provide insights into their characteristics.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors. This process is crucial when working with rational functions, as it simplifies the process of partial fraction decomposition by breaking down the denominator into simpler linear or quadratic terms.

In our example, the denominator is \( x^3 - x^2 + 2x - 2 \). By grouping, we factored it as \( (x-1)(x^2+2) \). Factorization can involve:

• Recognizing common factors in terms.
• Applying algebraic identities like the difference of squares.
• Using polynomial division or synthetic division for higher-degree polynomials.

Efficient factorization helps in writing the rational function in a form that is compatible with decomposition methods, allowing for simpler terms and making the solving process more manageable.

System of Equations

A system of equations comprises multiple equations that share common variables. Solving such systems is essential in partial fraction decomposition because it determines the unknown coefficients of the decomposed fractions.

In our solution, the equations arising from equating coefficients are:

• \( A + B = 3 \)
• \( C - B = -2 \)
• \( 2A - C = 8 \)

These equations emerge from comparing coefficients of the rational function after substituting and expanding terms. The process involves:

• Substitution: Express variables in terms of other variables.
• Elimination: Cancel variables by adding or subtracting equations.
• Simplification: Solve for each variable step by step.

The goal is to find specific values for \( A \), \( B \), and \( C \) that satisfy all equations, which ultimately enables forming the final decomposed expression. Understanding how to maneuver through systems of equations is key to mastering partial fraction decomposition.