Problem 27 - College Algebra Essentials

Question

write the partial fraction decomposition of each rational expression. $$ \frac{x^{2}}{(x-1)^{2}(x+1)} $$

Step-by-Step Solution

Answer

The partial fraction decomposition of the given rational expression \(\frac{x^{2}}{(x-1)^{2}(x+1)}\) is \(\frac{1}{2(x-1)} - \frac{1}{2(x-1)^{2}}\).

Step 1: Express the rational function as sum of Partial Fractions

The given expression can be rewritten as:\[\frac{x^{2}}{(x-1)^{2}(x+1)} = \frac{A}{x-1} + \frac{B}{(x-1)^{2}} + \frac{C}{x+1}\]Here, A, B, and C are constants which we need to find.

Step 2: Equate coefficients

Clear the fraction by multiplying both sides by \((x-1)^2(x+1)\) to get:\[x^{2} = A(x-1)(x+1) + B(x+1) + C(x-1)^2\]This should hold true for all values of x. So, we can pick values that make factors zero to solve for A, B, and C.

Step 3: Solve for A

Let's first put \(x = 1\) in the equation. The right-hand side becomes 2A. So,\[A = \frac{1}{2}\]

Step 4: Solve for B

Next, let's put \(x = -1\) in the equation. The right-hand side becomes -2B. So,\[B = -\frac{1}{2}\]

Step 5: Solve for C

Finally, let's put \(x = 0\) in the equation. The right-hand side becomes -C. So,\[C = 0\]

Step 6: Substitute the values of A, B, and C

The partial fraction decomposition of the given rational expression is thus:\[\frac{x^{2}}{(x-1)^{2}(x+1)} = \frac{1}{2(x-1)} - \frac{1}{2(x-1)^{2}}\]

Key Concepts

Rational Expressions

Rational expressions are quotients of two polynomials. Similar in form to a fraction, they have a numerator and a denominator. In mathematics, rational expressions are often simplified or restructured to aid in further calculations or analyses.

Understanding rational expressions involves recognizing that they can have undefined points, where the denominator equals zero. For example, in the expression \( \frac{x^2}{(x-1)^2(x+1)} \), the expression is undefined at \( x = 1 \) and \( x = -1 \) because these values make the denominator zero.

When working with rational expressions, always remember:

• Identify the domain by setting the denominator not equal to zero.
• Simplify the expression by factoring polynomials when possible.
• Rewriting complex rational expressions can simplify analysis, particularly in integrals or other algebraic operations.

These steps help in effectively managing calculations involving rational expressions, especially in calculus and algebra.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a complex rational expression as a sum of simpler fractions. This process is especially helpful in integration, allowing us to break down complicated expressions into manageable parts.

The key concept is to write a rational expression, such as \( \frac{x^2}{(x-1)^2(x+1)} \), as a sum of fractions whose denominators are the factors of the original denominator. For example, the decomposition may look like this: \[ \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{x+1} \]Each of these fractions has a polynomial numerator, often represented by constants (e.g., \(A\), \(B\), \(C\)).

Steps for decomposition include:

• Express the rational expression as a sum of partial fractions, assigning constants to the numerators.
• Clear the fraction by multiplying through by the common denominator.
• Determine the values of these constants by using substitution or equating coefficients.

This method makes solving integrals and differential equations more straightforward by simplifying the complexity of rational expressions.

Algebraic Techniques

Involved in partial fraction decomposition are essential algebraic techniques that simplify computations and solve equations efficiently.

One such technique includes polynomial long division, used when the degree of the numerator is greater than or equal to the degree of the denominator. Here, it's important to reduce the expression first before starting the decomposition.

Another vital skill is factoring, where the denominator is broken down into its simplest polynomial factors. This step often involves techniques such as recognizing common factors, using the quadratic formula, or polynomial identities.

Additionally, equating coefficients is crucial in partial fraction decomposition. This involves setting corresponding coefficients of expressions equal to each other to solve for the unknown constants:

• Multiply both sides by the denominator to remove fractions.
• Select strategic values for \(x\) to simplify solving for constants.
• Use substitution to isolate and determine each constant one by one.

These algebraic techniques allow mathematicians and students alike to manipulate and understand complex expressions with greater ease.