Problem 48 - Precalculus: Mathematics for Calculus

Question

Assembling and Disassembling Partial Fractions The following expression is a partial fraction decomposition. $$\frac{2}{x-1}+\frac{1}{(x-1)^{2}}+\frac{1}{x+1}$$ Use a common denominator to combine the terms into one fraction. Then use the techniques of this section to find its partial fraction decomposition. Did you get back the original expression?

Step-by-Step Solution

Answer

Yes, the original expression is recovered after recomposing and decomposing.

Step 1: Identify the Denominators

Identify the different denominators in the given partial fractions: \(x - 1\), \((x - 1)^2\), and \(x + 1\).

Step 2: Find the Common Denominator

The common denominator of the fractions \(\frac{2}{x-1}+\frac{1}{(x-1)^{2}}+\frac{1}{x+1}\) is \((x-1)^{2}(x+1)\).

Step 3: Rewrite Each Fraction with the Common Denominator

Each term needs to be rewritten over the common denominator: - \(\frac{2}{x-1} = \frac{2(x-1)(x+1)}{(x-1)^{2}(x+1)} = \frac{2(x+1)}{(x-1)^2(x+1)}\) - \(\frac{1}{(x-1)^{2}} = \frac{1(x+1)}{(x-1)^{2}(x+1)}\) - \(\frac{1}{x+1} = \frac{1(x-1)^2}{(x-1)^{2}(x+1)}\).

Step 4: Combine the Fractions

Add the fractions to form a single fraction: \[ \frac{2(x+1) + 1(x+1) + 1(x-1)^2}{(x-1)^2(x+1)} \].Simplifying the numerator gives: \[ \frac{2x + 2 + x + 1 + x^2 - 2x + 1}{(x-1)^2(x+1)} = \frac{x^2 + x + 4}{(x-1)^2(x+1)} \].

Step 5: Decompose the Combined Fraction

To find the partial fraction decomposition of \( \frac{x^2 + x + 4}{(x-1)^2(x+1)} \), assume:\[ \frac{x^2 + x + 4}{(x-1)^2(x+1)} = \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{x+1} \]Multiply both sides by \((x-1)^2(x+1)\) to clear the denominators:\[ x^2 + x + 4 = A(x-1)(x+1) + B(x+1) + C(x-1)^2 \].

Step 6: Solve for Coefficients A, B, and C

Expand and equate coefficients from both sides of the equation:- Constant term gives B = 1.- Term with \(x\) gives A = 2.- Term with \(x^2\) gives C = 1.Hence, the decomposition is \( \frac{2}{x-1} + \frac{1}{(x-1)^2} + \frac{1}{x+1} \).

Step 7: Compare with the Original Expression

Verify whether the decomposed fractions match the original expression. The decomposition is "\( \frac{2}{x-1} + \frac{1}{(x-1)^2} + \frac{1}{x+1}\)" which matches the original expression exactly.

Key Concepts

Common Denominator

When dealing with fractions, especially in algebra, the concept of a common denominator is crucial. It's important when you need to add, subtract, or compare fractions. In the exercise, each term in the partial fraction expression needs a common denominator to combine into a single fraction. The original fractions have different denominators: \(x-1\), \((x-1)^2\), and \(x+1\). To find a common denominator, you take the least common multiple (LCM) of these expressions.

The LCM for these terms is \((x-1)^2(x+1)\). This is because \((x-1)^2\) is the highest power of \(x-1\), and \(x+1\) is included to ensure all denominators can divide the common denominator without leaving a remainder. Using a common denominator helps in simplifying the algebraic expressions and is a foundational step in many algebraic operations, including fraction decomposition.

Fraction Decomposition

Fraction decomposition, or partial fractions, is the process of breaking down a complex fraction into simpler ones. This is particularly helpful when you want to work with or integrate rational expressions. After finding a combined fraction using a common denominator, you might want to break it down again into its components—this is where decomposition comes into play.

To decompose a fraction, you express the complex fraction as a sum of simpler fractions. In the exercise, the combined fraction \(\frac{x^2 + x + 4}{(x-1)^2(x+1)}\) is expressed as \(\frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{x+1}\). Here, \(A\), \(B\), and \(C\) are constants that need to be determined.

• Clear the denominators to compare coefficients by multiplying through the entire equation by the common denominator \((x-1)^2(x+1)\).
• Then, expand both sides and equate coefficients for corresponding powers of \(x\) to find the values of \(A\), \(B\), and \(C\).

This method allows you to easily revert to the original simpler fractions from a single complex fraction.

Algebraic Expressions

Algebraic expressions are fundamental in math as they allow us to represent numbers and quantities in equations and formulas. An algebraic expression involves variables, numbers, and arithmetic operations. In dealing with algebra, understanding how to manipulate these expressions is critical.

In this exercise, we deal with expressions structured in fractional form, where the numerator and the denominator are also algebraic expressions. The expression \(\frac{x^2 + x + 4}{(x-1)^2(x+1)}\) consists of a polynomial in the numerator and a factored polynomial in the denominator.

• Manipulating algebraic expressions requires understanding the rules of algebra, such as how to factor, expand, and simplify terms.
• In our example, expanding \(A(x-1)(x+1) + B(x+1) + C(x-1)^2\) involves applying the distributive property to both expand and simplify.

By knowing how to handle these steps smoothly, you can solve for different coefficients in partial fractions and get back to the simpler expressions, understanding both the structure and the meaning behind each component of the fraction.