Problem 40 - College Algebra

Question

Write the partial fraction decomposition of each rational expression. $$\frac{3 x^{3}-6 x^{2}+7 x-2}{\left(x^{2}-2 x+2\right)^{2}}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of \( \frac{3 x^{3}-6 x^{2}+7 x-2}{(x^{2}-2 x+2)^{2}} \) is \( \frac{A}{x^{2}-2x+2} + \frac{Bx+C}{(x^{2}-2x+2)^{2}} \) where A, B, and C are constants obtained by solving the resulting system of equations.

Step 1: Identify the factors of the denominator

The denominator is \( (x^{2}-2x+2)^{2} \), thus the factor is \( (x^{2}-2x+2) \).

Step 2: Set up the partial fraction decomposition

We express the rational function as the sum of two terms: \[ \frac{3 x^{3}-6 x^{2}+7 x-2}{(x^{2}-2 x+2)^{2}} = \frac{A}{x^{2}-2x+2} + \frac{Bx+C}{(x^{2}-2x+2)^{2}}\]

Step 3: Solve for A, B, and C

To solve for A, B, and C, multiply through by the common denominator \( (x^{2} - 2x + 2)^{2} \) to cancel out the denominators: \[ 3x^{3} - 6x^{2} + 7x - 2 = A(x^{2} - 2x + 2) + (Bx + C)(x^{2} - 2x + 2) \] Then, choose suitable values for x, or equate coefficients and solve for A, B, C.

Step 4: Formalize the result

The final decomposition will look like \[ \frac{3 x^{3}-6 x^{2}+7 x-2}{(x^{2}-2 x+2)^{2}} = \frac{A}{x^{2}-2x+2} + \frac{Bx+C}{(x^{2}-2x+2)^{2}}\] where A, B and C are the obtained values from the previous step.

Key Concepts

Rational Expressions

Rational expressions are similar to fractions but involve polynomials in the numerator and the denominator. They express a division of two algebraic expressions. Given the polynomial numerator and denominator, understanding their properties helps in simplifying and analyzing the expression.

To work with rational expressions, remember these key points:

• Simplification: Like numerical fractions, cancel common factors in the numerator and the denominator to simplify the expression.
• Operations: Addition, subtraction, multiplication, and division follow similar rules as those with fractions. Ensure the denominators are the same when adding or subtracting.
• Non-zero denominator: The values that make the denominator zero are excluded from the expression's domain to avoid division by zero.

In partial fraction decomposition, the goal is to break a complicated rational expression into a simpler sum of fractions that are easier to integrate or analyze. For example, you are given a rational expression with a polynomial of degree three in the numerator and a factored square polynomial in the denominator. Decomposing it into partial fractions helps solve integrals involving the rational expression.

Algebraic Fractions

Algebraic fractions involve variables in the numerator, the denominator, or both. They obey the same rules as numerical fractions. Understanding how to manipulate these expressions is vital for solving equations, simplifying expressions, and approaching problems like partial fraction decomposition.

Key aspects of dealing with algebraic fractions are:

• Factorization: Factor both the numerator and the denominator fully to reveal any cancelable factors.
• Finding a Common Denominator: When adding or subtracting algebraic fractions, convert them into equivalent fractions that share a common denominator.
• Proportions and Cross-Multiplication: Often used to simplify more complex expressions or solve for variables within the expression.

When doing partial fraction decomposition, express a complex algebraic fraction into simpler individual fractions. This decomposition makes integration, among other operations, much more manageable by handling simpler parts.

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another. When dealing with rational expressions, this technique can simplify expressions or solve problems involving partial fraction decompositions.

When to use polynomial long division:

• Degree Consideration: Use it when the degree of the numerator is higher than that of the denominator. This will help reduce the degree before attempting partial fractions.
• Result Form: The division provides a quotient and a remainder, allowing further simplification of the rational expression or setting it up for partial fraction decomposition.

If the numerator has a higher degree than the factors of the denominator in the partial fraction task, use polynomial long division first to simplify the expression before breaking down the terms. When performing long division, write the dividend under the long division symbol and divide by the divisor step-by-step until the remainder is less than the degree of the divisor. This step makes subsequent algebraic manipulations more straightforward.