Problem 45 - Precalculus: Mathematical for Calculus

Question

Recognizing Partial Fraction Decompositions For each expression, determine whether it is already a partial fraction decomposition, or whether it can be decomposed further. (a) \(\frac{x}{x^{2}+1}+\frac{1}{x+1}\) (b) \(\frac{x}{(x+1)^{2}}\) (c) \(\frac{1}{x+1}+\frac{2}{(x+1)^{2}}\) (d) \(\frac{x+2}{\left(x^{2}+1\right)^{2}}\)

Step-by-Step Solution

Answer

(a) already a decomposition, (b) can be decomposed, (c) already a decomposition, (d) can be decomposed further.

Step 1: Identify the Type of Each Denominator

Examine denominators of each expression to determine if they correspond to linear or irreducible quadratic factors. This is critical for identifying possible partial fraction decompositions.

Step 2: Analyze Expression (a)

The expression \( \frac{x}{x^2+1} + \frac{1}{x+1} \) combines a term with an irreducible quadratic denominator and a linear denominator. This is already in partial fraction form because the denominator terms are distinct and not repeated.

Step 3: Evaluate Expression (b)

The fraction \( \frac{x}{(x+1)^2} \) has a squared linear factor. Therefore, it can be decomposed into partial fractions as \( \frac{A}{x+1} + \frac{B}{(x+1)^2} \) with constants \( A \) and \( B \) to be determined.

Step 4: Consider Expression (c)

In \( \frac{1}{x+1} + \frac{2}{(x+1)^2} \), the terms are based on repeated linear factors \((x+1)\). Each term already represents a part of a partial fraction decomposition following the symmetric model \( \frac{A}{x+1} + \frac{B}{(x+1)^2} \), so it cannot be decomposed further.

Step 5: Inspect Expression (d)

The expression \( \frac{x+2}{(x^2+1)^2} \) involves a repeated irreducible quadratic denominator. It can be further decomposed as \( \frac{Ax+B}{x^2+1} + \frac{Cx+D}{(x^2+1)^2} \), where \( A, B, C, \) and \( D \) are constants to find. This is necessary to express as partial fractions.

Key Concepts

Linear Factors

Linear factors are expressions formed by polynomials of degree one, basically equations in the form of \( ax + b \). These are straightforward and quite common when dealing with partial fraction decomposition, as they are easily recognizable and simplified.

When performing partial fraction decomposition with linear factors, we break a fraction down into simpler fractions whose denominators are linear expressions. For instance, if you have \( \frac{x}{(x+1)^2} \) as in expression (b), you identify the linear factor, which in this case is \( x+1 \). Because it is squared, we express it using partial fractions, writing it as \( \frac{A}{x+1} + \frac{B}{(x+1)^2} \) where \( A \) and \( B \) are constants we need to solve for. The aim is to simplify computations and integration processes by dealing with simple terms.

Irreducible Quadratic Factors

Irreducible quadratic factors are polynomials of degree two that cannot be factored further using real numbers. These typically appear as expressions like \( x^2 + 1 \) that don't have roots in the real number system.

In partial fraction decomposition, when you encounter an irreducible quadratic factor, like in expression (a) \( \frac{x}{x^2+1} + \frac{1}{x+1} \), you cannot break it down further using simpler real-number denominators. Instead, you acknowledge it as already in partial fraction form if the irreducible quadratic factor is not repeated and coupled with linear terms. This helps in achieving a simpler expression that is more manageable in calculus and higher algebra applications.

Repeated Factors

Repeated factors occur when a factor is raised to a power greater than one in the denominator. This requires special handling in partial fraction decomposition because each power of the factor needs its own term.

For instance, in expression (c) \( \frac{1}{x+1} + \frac{2}{(x+1)^2} \), the repeated factor \( x+1 \) appears twice with different powers. This setup already matches the partial fraction decomposition for repeated linear factors, showing no further breakdown is needed. However, if you encounter repeated irreducible quadratic factors, as in expression (d) \( \frac{x+2}{(x^2+1)^2} \), you decompose it into \( \frac{Ax+B}{x^2+1} + \frac{Cx+D}{(x^2+1)^2} \). Here, you ensure to assign unique coefficients for each term, reflecting the complexity of dealing with higher-order repeats.