Problem 47 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

(a) Decomposed, (b) Not decomposed, (c) Decomposed, (d) Not decomposed.

Step 1: Analyze Expression (a)

Expression (a) is \( \frac{x}{x^2+1} + \frac{1}{x+1} \). Check if it satisfies the conditions for a partial fraction decomposition. It already consists of two separate fractions with linear or irreducible quadratic denominators. Hence, it is already in partial fraction decomposition form.

Step 2: Analyze Expression (b)

Expression (b) is \( \frac{x}{(x+1)^2} \). The denominator is \((x+1)^2\), which suggests that decomposition could involve fractions like \( \frac{A}{x+1} + \frac{B}{(x+1)^2} \). This expression can be decomposed further as it does not yet reflect such a form.

Step 3: Analyze Expression (c)

Expression (c) is \( \frac{1}{x+1} + \frac{2}{(x+1)^2} \). It consists of two fractions with \(x+1\) and \((x+1)^2\) as denominators, fitting the partial fraction decomposition format already. Hence, this one does not require further decomposition.

Step 4: Analyze Expression (d)

Expression (d) is \( \frac{x+2}{(x^2+1)^2} \). The denominator \((x^2+1)^2\) is a combination that can be broken down into a form incorporating terms with \((x^2+1)\) and \((x^2+1)^2\). For a proper decomposition, we should express it as \( \frac{Ax+B}{x^2+1} + \frac{Cx+D}{(x^2+1)^2} \). Since it can take this form, expression (d) can be decomposed further.

Key Concepts

Rational Expressions

Rational expressions are mathematical expressions involving ratios of polynomials. In simpler terms, they are fractions where the numerator and denominator are both polynomials. Understanding rational expressions is crucial because they form the foundation for solving many algebraic problems. A key characteristic of rational expressions is that they can be simplified, manipulated, or decomposed using various algebraic techniques. This includes factoring, canceling common factors, and applying partial fraction decomposition.

• Numerator: The polynom that's positioned above the fraction line.
• Denominator: The polynom that's positioned below the fraction line. It must not equal zero because division by zero is undefined.

Learning how to handle rational expressions also allows students to better understand complex mathematical concepts, such as limits, integrals, and differential equations, often used in calculus and higher mathematics.

Algebraic Fractions

Algebraic fractions are simply fractions that contain algebraic expressions in the numerator, the denominator, or both. They are similar to simple numerical fractions, but instead of numbers, they have expressions that can include variables. The ability to work with algebraic fractions is essential, as it builds a strong algebra foundation necessary for advanced topics. When dealing with these fractions, it's important to know how to:

• Simplify: Combine like terms and ensure that the expression is in the smallest form possible.
• Multiply and Divide: Follow similar steps as with numerical fractions, ensuring the multiplication and division of both numerators and denominators.
• Add and Subtract: Find a common denominator when necessary, then add or subtract the numerators.

Understanding algebraic fractions allows one to easily transition to more complex algebraic concepts and operations, including partial fraction decomposition, which often involves breaking down complex algebraic fractions into simpler parts.

Irreducible Quadratic Denominators

An irreducible quadratic denominator is a quadratic expression in the denominator of a fraction that cannot be factored further over the real numbers. For example, an expression like \(x^2 + 1\) is irreducible over the real numbers because it has no real roots.Irreducible quadratic denominators often appear in the partial fraction decomposition process. When you see an irreducible quadratic denominator, it indicates that the partial fraction form will include linear terms in the numerator of the decomposed fractions.

• Example: For a fraction with an irreducible quadratic denominator \(x^2 + 1\), the partial fraction would take the form \( \frac{Ax + B}{x^2 + 1} \).
• Significance: Recognizing these irreducible components is essential for the accurate decomposition of algebraic fractions and for solving corresponding equations.

Understanding how to handle irreducible quadratic denominators helps in not only partial fraction decomposition but also in identifying and solving complex rational expressions in calculus and algebra.