Problem 39 - College Algebra

Question

Write the partial fraction decomposition of each rational expression. $$\frac{x^{3}-4 x^{2}+9 x-5}{\left(x^{2}-2 x+3\right)^{2}}$$

Step-by-Step Solution

Answer

The final answer for the partial fraction decomposition will be obtained by placing the constants A, B, C, and D from Step 4 into the equation from Step 2. The corresponding equation represents the decomposition of the original rational expression into partial fractions.

Step 1: Factor the denominator

The first step is to factorize the denominator expression \((x^{2}-2 x+3)^{2}\). However, this quadratic has no real roots, and it can't be factored in terms of real numbers. Instead, it can be expressed as \((x -1 + i\sqrt{2})^{2}(x -1 - i\sqrt{2})^{2}\). So the complex numbers \(1 + i\sqrt{2}\) and \(1 - i\sqrt{2}\) are the roots of the denominator.

Step 2: Formulate the general form of the partial decomposition

Once the roots of the denominator are found, the rational function can be broken down as: \[ \frac{x^{3}-4 x^{2}+9 x-5}{(x -1 + i\sqrt{2})^{2}(x -1 - i\sqrt{2})^{2}} = \frac{A}{x -1 + i\sqrt{2}} + \frac{B}{(x -1 + i\sqrt{2})^{2}} + \frac{C}{x -1 - i\sqrt{2}} + \frac{D}{(x -1 - i\sqrt{2})^{2}}\] Where A, B, C, and D are constants to be determined.

Step 3: Clear the fractions and collect like terms

Multiply both sides of the equation by the denominator on the left to clear the fractions and simplify to collect like terms. Equate the coefficients on both sides of the result to obtain a system of equations in terms of A, B, C, and D.

Step 4: Solve the system of equations

Solving the system of equation will lead to the values of A, B, C, and D. They will provide the coefficients for the partial fraction decomposition.

Step 5: Write the final answer

Place the values of A, B, C, and D into the equation from Step 2 to write the final partial fraction decomposition.

Key Concepts

Rational Expressions

Rational expressions are fractions that include polynomials in their numerator and/or denominator. The process of partial fraction decomposition involves breaking down these complex fractions into simpler fractions. This method is particularly useful for integrating rational functions or simplifying them for other mathematical purposes.

When working with rational expressions, it's important to ensure that:

• The expression is simplified—factor the numerator and denominator when possible.
• The denominator is not zero, as division by zero is undefined.

Before performing partial fraction decomposition, consider checking for any immediate simplifications that can be made to the rational expression first.

Complex Roots

Complex roots arise when a quadratic expression does not have real solutions. This usually happens when the discriminant (\(b^2 - 4ac\)) of the quadratic is negative. In such cases, the roots are complex, and they come in conjugate pairs. For the exercise, the quadratic \(x^2 - 2x + 3\) has a negative discriminant, indicating complex roots.

These roots can be found using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Utilizing this method, the exercise yields roots of \(1 + i\sqrt{2}\) and \(1 - i\sqrt{2}\). Complex roots add an additional layer to partial fraction decomposition, requiring a careful approach to forming equivalent expressions with simpler terms. This is vital for simplifying rational expressions that involve complex numbers.

• Identify the quadratic component in the denominator.
• Check the discriminant to determine if roots are real or complex.
• If complex, express the denominator using these complex conjugates.

System of Equations

A system of equations means solving multiple equations simultaneously, where each equation may involve the same set of unknowns. In the context of partial fraction decomposition, after establishing the general form of the decomposition, you need to solve for unknown constants.

Here's how to approach it:

• Substitute a common denominator and cross-multiply to clear fractions.
• Expand and simplify both sides of the equation.
• Equate the coefficients of like powers of the variable (typically x).

This equating gives rise to a system of linear equations. By solving these linear equations simultaneously, you determine the constants that serve as coefficients in your partial fraction.

Methods for solving these systems include substitution, elimination, or using matrix operations if dealing with more complex systems.

Factoring Polynomials

Factoring is the process of finding numbers or expressions that multiply together to make a given expression or number. In the context of rational expressions, you are typically looking to break down the denominator into its simplest components.

Factoring polynomials can be done by:

• Identifying common factors;
• Using the difference of squares, trinomial squares, or cubic forms;
• Applying synthetic division or other techniques if direct factoring isn’t possible.

For the exercise at hand, despite attempting to factor the quadratic \(x^2 - 2x + 3\), it turned out to be non-factorable over the reals because of its complex roots. Factoring here then entails recognizing the polynomial factors in terms of complex conjugates. While factoring helps simplify numerous algebraic operations, in cases such as these where complex roots exist, it often necessitates expressing them in forms conducive to further manipulation, such as during partial fraction decomposition.