Problem 47 - Precalculus: Mathematics 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 decomposed; (b) Can be decomposed further; (c) Already decomposed; (d) Can be decomposed further.

Step 1: Understand Partial Fraction Decomposition

Partial fraction decomposition is a way of expressing a fraction as a sum of fractions with simpler denominators. For this, the denominator should not be repeated or the factor should be irreducible quadratic for further decomposition. Let's apply this to each case.

Step 2: Analyze Expression (a)

Expression: \(\frac{x}{x^{2}+1}+\frac{1}{x+1}\).The denominators \(x^{2}+1\) and \(x+1\) are distinct and neither are further factorable in terms of real numbers. Thus, this expression is already in partial fraction decomposition form.

Step 3: Analyze Expression (b)

Expression: \(\frac{x}{(x+1)^{2}}\).This can be potentially decomposed further. We need to express it as: \(\frac{A}{x+1}+\frac{B}{(x+1)^{2}}\). This needs further decomposition.

Step 4: Analyze Expression (c)

Expression: \(\frac{1}{x+1}+\frac{2}{(x+1)^{2}}\).This is expressed with distinct powers of the same linear factor \((x+1)\) as separate terms. It is already in partial fraction decomposition form.

Step 5: Analyze Expression (d)

Expression: \(\frac{x+2}{(x^{2}+1)^{2}}\).The term \((x^{2}+1)\) is irreducible over real numbers, but there is a potential to decompose using the fact that \((x^{2}+1)\) is squared: \(\frac{Ax+B}{x^{2}+1} + \frac{Cx+D}{(x^{2}+1)^{2}}\). This expression can be decomposed further.

Key Concepts

Fraction Decomposition

Fraction decomposition is a process used to break down complex algebraic fractions into simpler parts. This is especially useful when dealing with fractional expressions that are difficult to integrate or simplify directly. The central idea of fraction decomposition is to take a complicated rational expression and express it as the sum of smaller, simpler fractions. Each portion has a denominator that is a factor of the original fraction's denominator. This method is powerful in making integrals and other operations more manageable.For example, consider the expression \[ \frac{x}{(x+1)^2} \] To decompose it, we rewrite the expression as a sum: \[ \frac{A}{x+1} + \frac{B}{(x+1)^2} \] We then determine the constants \( A \) and \( B \) by equating and solving equations.
This approach creates terms that integrate easily, supporting deeper mathematical analysis.

Rational Expressions

Rational expressions are essentially fractions where both the numerator and the denominator are polynomials. They play a crucial role in algebra, calculus, and beyond.A rational expression is in its simplest form when:

• The greatest common divisor (GCD) of the numerator and denominator is 1.
• All the polynomial terms are reduced as much as possible.

When working with rational expressions, common tasks include simplifying, performing arithmetic operations, and solving equations.Consider \[ \frac{1}{x+1} + \frac{2}{(x+1)^2} \] This expression consists of two rational expressions with shared polynomial base \((x+1)\). Each fraction represents part of a larger decomposition.
Operations with these expressions require understanding each fraction's unique contribution to the overall sum.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both are algebraic expressions. They are significant in algebra because they often need simplifying or decomposition to solve equations or analyze functions easily.When working with algebraic fractions:

• Ensure all expressions are simplified to their smallest form.
• Factor numerators and denominators to find common terms.
• Perform partial fraction decomposition when necessary to break down complex expressions into easier components.

Consider the expression \[ \frac{x+2}{(x^2+1)^2} \] The fraction's denominator is a polynomial raised to a power, suggesting that we can decompose the expression further:\[ \frac{Ax+B}{x^2+1} + \frac{Cx+D}{(x^2+1)^2} \] This decomposition separates terms for easier analysis, highlighting algebraic fractions' ability to transform complex problems into manageable parts.