Problem 2 - Algebra and Trigonometry

Question

\(1-2\) . For each rational function \(r,\) choose from (i) \(-(\text { iv ) the }\) appropriate form for its partial fraction decomposition. $$ \begin{array}{l}{r(x)=\frac{2 x+8}{(x-1)\left(x^{2}+4\right)}} \\ {\text { (i) } \frac{A}{x-1}+\frac{B}{x^{2}+4}} \\ {\text { (ii) } \frac{A}{x-1}+\frac{B x+C}{x^{2}+4}} \\ {\text { (iii) } \frac{A}{x-1}+\frac{B}{x+2}+\frac{C}{x^{2}+4}} \\ {\text { (iv) } \frac{A x+B}{x-1}+\frac{C x+D}{x^{2}+4}}\end{array} $$

Step-by-Step Solution

Answer

The correct partial fraction decomposition form is option (ii).

Step 1: Analyze the Denominator

The denominator of the given rational function is \((x-1)(x^2+4)\). This is a product of a linear factor \((x-1)\) and an irreducible quadratic factor \((x^2 + 4)\). We'll use this information to determine the correct form for partial fraction decomposition.

Step 2: Determine the Correct Decomposition Form

For a rational function with a linear factor \((x-1)\) and an irreducible quadratic factor \((x^2+4)\), the appropriate form of the partial fraction decomposition includes a separate term for each part: a constant over the linear term and a linear expression over the quadratic term. Therefore, the appropriate choice is option (ii): \[ \frac{A}{x-1} + \frac{Bx+C}{x^2+4} \]

Step 3: Verification

Verify the selected form fits the structure of the partial fraction decomposition. The presence of a linear term (A) over \((x-1)\) and a linear numerator (Bx + C) over the quadratic \((x^2 + 4)\) confirms option (ii) is correct. No other options match this decomposition matching linearly and quadratically, confirming (ii) is the right choice.

Key Concepts

Rational Functions

Rational functions are mathematical expressions represented as the ratio of two polynomials. They take the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). These functions are central to algebra, as they appear in various mathematical applications and calculus problems.

• The numerator \( P(x) \) determines the degree of the polynomial on top.
• The denominator \( Q(x) \) plays a crucial role since it must never be zero to maintain the function's validity.

Partial fraction decomposition is a technique to simplify rational functions. It is particularly useful when integrating complex rational expressions. By breaking them down into simpler parts, we can manage the calculations more efficiently.

Linear Factors

In the context of rational functions, a linear factor appears as a first-degree polynomial of the form \( (x-a) \). These factors are straightforward and can be solved with basic algebraic techniques.

• A factor like \( (x-1) \) impacts the form of partial fraction decomposition because it allows us to express part of the function as \( \frac{A}{x-1} \).
• Linear factors emerge when a polynomial equation equals zero, giving us the roots.

Identifying these factors enables us to understand more about the function's behavior and roots. They are fundamental in constructing and simplifying rational expressions.

Quadratic Factors

Quadratic factors are irreducible polynomials of the form \( ax^2 + bx + c \) where \( a eq 0 \). When they cannot be factored further over the real numbers, we use them in their existing quadratic form.

• A common example, like \( (x^2+4) \), cannot be factored into linear components using real numbers.
• In partial fraction decomposition, this factor is represented with a linear numerator over the quadratic factor, as \( \frac{Bx+C}{x^2+4} \).

Quadratic factors, when involved in expressions, require additional steps for simplification, such as long division or completing the square. They often complicate the decomposition and require algebraic techniques to solve.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. It helps us represent complex problems in an understandable way using expressions, equations, and functions. The foundation of algebra lies in understanding how to rearrange and solve equations.

• Partial fraction decomposition is an algebraic method used to break down complex rational expressions into simpler ones.
• Using algebra, we can manipulate and simplify expressions with both linear and quadratic factors, providing exact points for solving equations.

Algebra plays a pivotal role when tackling rational functions, as it enables the determination of relationships and numerical solutions, especially through techniques like partial fraction decomposition. By mastering these algebraic methods, solving complex mathematical challenges becomes feasible and efficient.