Problem 63 - Algebra and Trigonometry

Question

determine whether each statement makes sense or does not make sense, and explain your reasoning. Because \((x+3)^{2}\) consists of two factors of \(x+3,1\) set up the following partial fraction decomposition: $$\frac{5 x+2}{(x+3)^{2}}=\frac{A}{x+3}+\frac{B}{x+3}$$

Step-by-Step Solution

Answer

The given statement expressing the partial fractions is not correct. The correct expression should be \(\frac{5 x+2}{(x+3)^{2}}=\frac{A}{x+3}+\frac{B}{(x+3)^2}\) as the degree of the denominator in each factor should be one less than the original fraction.

Step 1: Understanding of Partial Fraction Decomposition

Partial fraction decomposition is a method where a fraction whose denominator is a polynomial, gets expressed as a sum of fractions with simpler denominators. It involves expressing the fraction as the sum of a polynomial (possibly zero) and a sum of fractions, where each fraction has a simpler denominator. This can facilitate further computations regarding the fraction.

Step 2: Apply the Concept to Validate Statement

We apply the concept to the given fraction \(\frac{5 x+2}{(x+3)^{2}}\). You have two identical factors in the denominator i.e., (x+3). But, instead of one single constant divided by each factor, there should be one degree less than the original factor in the numerator. So, the correct form of partial fraction decomposition should be \(\frac{5 x+2}{(x+3)^{2}}=\frac{A}{x+3}+\frac{B}{(x+3)^2}\) where A and B are constants.

Step 3: Reasoning of the Validity

It can be observed that the decomposition presented in the statement \(\frac{5 x+2}{(x+3)^{2}}=\frac{A}{x+3}+\frac{B}{x+3}\) does not make sense because it contradicts the proper method of decomposition. As per the fundamental rule of partial fractions, the numerator’s degree should be one less than the denominator. Hence, the second fraction must have the denominator as \((x+3)^2\) and not \(x+3\).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It provides a way to describe and analyze relationships between varying quantities mathematically. When it comes to partial fraction decomposition, algebra plays a critical role as it involves breaking down complex rational expressions into simpler, more manageable ones.

Through the lens of algebra, partial fraction decomposition transforms a single complex fraction into a sum of simpler fractions, aiding in integrations and solving equations. This approach is particularly useful when dealing with integrals in calculus or finding inverse Laplace transforms in engineering.

Polynomial Denominator

When we speak of a polynomial denominator, we're referring to the bottom part of a rational expression that is a polynomial. A polynomial is an algebraic expression made up of constants, variables (also known as indeterminates), and positive integer exponents of those variables, usually combined using addition, subtraction, and multiplication.

The order of the polynomial is determined by the highest exponent of the variables. In the context of partial fraction decomposition, each term in the denominator suggests a potential component in the final expression. Specially, when a polynomial denominator can be factored into linear or quadratic factors, each of which may be repeated, we apply partial fraction decomposition to break the original expression into parts that are easier to work with. Understanding the structure of polynomial denominators is key to mastering this technique.

Rational Expressions

Rational expressions are fractions in which both the numerator and the denominator are polynomials. These expressions play a significant role in various fields of mathematics, including algebra and calculus, because they encapsulate the notion of division and ratio within algebraic contexts.

Partial fraction decomposition is a method specifically tailored to handle rational expressions whose denominators are polynomials that can be factored into simpler polynomials. The aim is to represent the original rational expression as a combination of simpler ones whose denominators are these factors. This becomes incredibly valuable while attempting to perform operations such as integration or simplification, where working with the original complex fraction might be challenging. Having a firm grasp of how rational expressions operate is crucial for properly applying the partial fraction decomposition technique.