Problem 4 - Thomas Calculus

Question

Expand the quotients in Exercises \(1-8\) by partial fractions. $$\frac{2 x+2}{x^{2}-2 x+1}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Type of Partial Fraction Decomposition

The given function is a fraction \( \frac{2x+2}{x^2-2x+1} \). First, observe that the denominator \( x^2-2x+1 \) is a perfect square, specifically \( (x-1)^2 \). Since the denominator is a quadratic repeated factor, the partial fraction decomposition will be of the form \( \frac{A}{x-1} + \frac{B}{(x-1)^2} \).

Step 2: Setup the Partial Fraction Equation

Set up the partial fraction equation by writing: \( \frac{2x+2}{x^2-2x+1} = \frac{A}{x-1} + \frac{B}{(x-1)^2} \).

Step 3: Eliminate the Denominator

Multiply both sides by \( (x-1)^2 \) to eliminate the denominators: \( 2x+2 = A(x-1) + B \).

Step 4: Expand and Simplify the Equation

Expand the right side: \( A(x-1) + B = Ax - A + B \). Set this equal to the left side: \( 2x + 2 = Ax - A + B \).

Step 5: Compare Coefficients

By matching the coefficients of corresponding powers of \( x \), we have two equations:1. For \( x \): \( A = 2 \)2. For the constant term: \( -A + B = 2 \).

Step 6: Solve for Constants

We already found \( A = 2 \). Substitute \( A = 2 \) into the second equation: \( -2 + B = 2 \)\( B = 4 \).

Step 7: Write the Final Partial Fraction Decomposition

Substitute \( A \) and \( B \) back into the partial fraction form: \( \frac{2}{x-1} + \frac{4}{(x-1)^2} \).

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree 2. It has a general form of:\[ ax^2 + bx + c = 0 \]where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). Quadratic equations are widely used in various mathematical calculations and can be solved using methods such as factoring, completing the square, or using the quadratic formula.
For the given problem, the denominator \( x^2 - 2x + 1 \) is a quadratic expression. In this particular case, the quadratic expression is unique because it factors into a perfect square, \((x-1)^2\). This factorization helps simplify the problem and guides us to the correct form of partial fraction decomposition.

Perfect Square

A perfect square is a quadratic expression that can be expressed as the square of a binomial. For example, the expression \( (x-1)^2 \) is a perfect square, meaning:\[ (x-1)^2 = x^2 - 2x + 1 \]Identifying a perfect square in partial fraction decomposition is crucial, as it can change the approach you take to find the solution. Recognizing \( x^2 - 2x + 1 \) as \( (x-1)^2 \) simplifies setting up the decomposition since the factor \( x-1 \) is repeated.
Understanding and recognizing perfect squares helps in:

• Simplifying expressions
• Reducing algebraic complexity
• Guiding the correct setup of partial fraction solutions

Coefficient Matching

Coefficient matching is a technique used to solve equations by aligning coefficients of corresponding terms. In the context of partial fraction decomposition, it helps determine the unknowns or constants in the numerator. After expressing the original rational function in its decomposed form, the goal is to align both sides by matching the terms with the same power of \( x \).
In the example solution, multiplying through by \( (x-1)^2 \) results in:\[ 2x + 2 = Ax - A + B \]Here, the coefficients of \( x \) are matched to find \( A \), while constant terms are used to solve for \( B \). From the equation:

• Term for \( x \): \( A = 2 \)
• Constant term: \( -A + B = 2 \)

This systematic alignment simplifies solving for unknown constants, providing a clearer path to the final solution.

Rational Functions

A rational function is defined as the quotient of two polynomials. These functions have the form:\[ f(x) = \frac{P(x)}{Q(x)} \]where \( P(x) \) and \( Q(x) \) are polynomial expressions and \( Q(x) eq 0 \).
Rational functions are significant in many applications where relationships between quantities are non-linear.
In the given problem, \( \frac{2x+2}{x^2-2x+1} \) is a rational function, with a numerator \( 2x+2 \) and a denominator \( x^2-2x+1 \). Such functions might present challenges in integration and calculus, paving the way for techniques like partial fraction decomposition. This method facilitates simplification by expressing the function as a sum of simpler fractions, making integration or further analysis more manageable.