Problem 5 - Precalculus

Question

Match the rational expression \((1-6)\) with the form of the partial-fraction decomposition \((a-f)\). a. \(\frac{A}{x}+\frac{B}{x^{2}}+\frac{C x+D}{x^{2}+25}\) b. \(\frac{A}{x}+\frac{B x+C}{x^{2}+25}+\frac{D x+E}{\left(x^{2}+25\right)^{2}}\) c. \(\frac{A}{x}+\frac{B x+C}{x^{2}+25}\) d. \(\frac{A}{x}+\frac{B}{x+5}+\frac{C}{x-5}\) e. \(\frac{A}{x}+\frac{B}{x^{2}}+\frac{C x+D}{x^{2}+25}+\frac{E x+F}{\left(x^{2}+25\right)^{2}}\) f. \(\frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x+5}+\frac{D}{x-5}\) $$\frac{3 x+2}{x\left(x^{2}+25\right)^{2}}$$

Step-by-Step Solution

Answer

The correct partial fraction form is option (e).

Step 1: Analyzing the Denominator

The given rational expression is \( \frac{3x+2}{x(x^2+25)^2} \). The denominator is \( x(x^2+25)^2 \), which suggests the structure involves linear and repeated irreducible quadratic factors. We identify \( x \) as a linear factor, and \( (x^2+25) \) as an irreducible quadratic factor that is repeated.

Step 2: Identifying Appropriate Partial Fractions

Based on the factorization \( x(x^2+25)^2 \), the partial fraction decomposition must include a term for each distinct factor and additional terms for repeated factors. We need terms for the linear factor \( x \), the quadratic factor \( x^2+25 \), and its repetition. Following standard rules for such decompositions, we expect to see terms: \( \frac{A}{x} \), \( \frac{B}{x^2} \), \( \frac{Cx+D}{x^2+25} \), and \( \frac{Ex+F}{(x^2+25)^2} \).

Step 3: Matching with Given Options

We compare our predicted form \( \frac{A}{x} + \frac{B}{x^2} + \frac{Cx+D}{x^2+25} + \frac{Ex+F}{(x^2+25)^2} \) with the options provided. This form matches with option (e) precisely, as it includes all terms corresponding to each factor of the denominator, consistent with our expectations for repeated irreducible quadratics and linear factors.

Key Concepts

Understanding Rational Expressions

A rational expression is simply a fraction where the numerator and the denominator are both polynomials. It's like a ratio of two polynomials. The power of rational expressions comes from their ability to represent complex relationships in a simplified form.
Breaking down these expressions can help us in simplifying computations, especially in calculus and algebra. For example,

• In the expression \( \frac{3x+2}{x(x^2+25)^2} \), we have both a polynomial in the numerator and denominator.
• The decomposition of such expressions into simpler 'partial fractions' aids in calculating integrals, among other things.

Understanding them is fundamental in advanced mathematics, as they appear frequently in topics like systems of equations and algebraic fractions.

Irreducible Quadratic Factors

Irreducible quadratic factors are quadratics that cannot be factored any further using real numbers.
For a quadratic \( ax^2 + bx + c \) to be irreducible, the discriminant \( b^2 - 4ac \) must be negative. This ensures that there are no real roots.
A common example of this is \( x^2 + 25 \). Here,

• The discriminant \( 0^2 - 4(1)(25) = -100 \) is negative, proving it has no real roots.
• Because it can't be broken down further using real numbers, it remains in its quadratic form.

These factors are considered when performing partial fraction decomposition, as they introduce terms such as \( \frac{Cx + D}{x^2+25} \). These terms account for linear combinations that's critical for the integrity of the decomposition.

Repeated Factors in Decomposition

Repeated factors occur when a factor in the denominator appears more than once. In decomposition, this requires extra attention.
For instance, the decomposing denominator \( (x^2+25)^2 \) includes \( x^2+25 \) twice, hence it's repeated. This necessitates the inclusion of multiple terms to account for each repetition.
Specifically,

• The first appearance gives us a term \( \frac{Cx + D}{x^2+25} \).
• The second, repeated appearance requires an additional term \( \frac{Ex + F}{(x^2+25)^2} \).

By including extra terms for repeated factors, we ensure that any potential contributions to the rational expression's structure are fully captured in the model. This captures more nuances in the behavior of the expression, allowing for a more complete and accurate decomposition.

Decoding Linear Factors

Linear factors are typically the simplest type of factors involved in partial fraction decomposition. A linear factor has the form \(x + a\).
They are straightforward because they do not involve powers greater than one or complex components.
In our example,

• The term \( x \) is a linear factor.
• This requires us to include a term such as \( \frac{A}{x} \) in our decomposition.

Linear factors play a crucial role in simplifying complex expressions because they often appear as standalone entities in the denominator, leading to the simplest fractional terms. While their simplicity might make them seem less significant, they're foundational in constructing decompositions that reflect the underlying algebraic relationships of the rational expression.