Problem 10 - Precalculus with Limits

Question

Writing the Form of the Decomposition. Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$\frac{x-2}{x^{2}+4 x+3}$$

Step-by-Step Solution

Answer

The form of the partial fraction decomposition is \(\frac{A}{x+1} + \frac{B}{x+3}\)

Step 1: Factor the Denominator

First, the denominator of the original fraction needs to be factored. Here, \(x^{2}+4x+3\) can be factored into \((x+1)(x+3)\)

Step 2: Write the Decomposition Form

The form of the partial fraction decomposition of the rational expression is based on the factors of the denominator. For factors of the form \((x-a)\) the corresponding partial fraction is \(\frac{A}{x-a}\), where A is a constant. Hence, the form of the partial fraction decomposition for the given rational fraction will be \(\frac{A}{x+1} + \frac{B}{x+3}\) where A and B are constants to be determined.

Key Concepts

Factoring Polynomials

Factoring polynomials is akin to breaking down a number into its prime factors, but for algebraic expressions. It's fundamental in simplifying and solving equations as well as in performing operations such as partial fraction decomposition. A polynomial is an expression made up of variables and coefficients, combined using only addition, subtraction, multiplication, and non-negative integer exponents.

For instance, in our exercise \(x^{2}+4x+3\), we can factor it by looking for two numbers that multiply to give the constant term (3) and add up to the coefficient of the linear term (4). These numbers are 1 and 3, thus the factored form is \(x+1)(x+3)\).

Knowing how to factor polynomials is essential, not just for partial fraction decomposition, but also in other areas of algebra and calculus. It simplifies complex expressions and unveils solutions that might not be immediately obvious.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They resemble ratios of polynomials and hence the name, rational expressions. Understanding how to handle these expressions is crucial as they appear frequently in calculus, algebra, and real-world problems.

In the context of our exercise, \(\frac{x-2}{x^{2}+4x+3}\) is a rational expression with a numerator \(x-2\) and a factored denominator \(x+1)(x+3)\). Just like fractions with numbers, rational expressions can often be simplified by finding common factors in the numerator and denominator, although that's not applicable in this particular case.

Working with rational expressions often leads us to perform operations such as partial fraction decomposition, which simplifies the expression further, making calculus techniques like integration more manageable.

Algebraic Fractions

Algebraic fractions are just like the fractions you first learned about — except instead of integers, there are polynomials in the numerator, the denominator, or both. A key skill when dealing with algebraic fractions is the ability to decompose them into simpler parts when the denominator has several factors.

In partial fraction decomposition, an algebraic fraction like \(\frac{x-2}{x^{2}+4x+3}\), is expressed as the sum of simpler fractions, in this case, \(\frac{A}{x+1} + \frac{B}{x+3}\). This technique is incredibly helpful in integral calculus, as it converts difficult-to-integrate expressions into simpler ones that can be handled more directly.

Understanding algebraic fractions and how to decompose them also aids in grasping the relationships between polynomial roots and their coefficients, and it plays a significant role in solving higher degree polynomial equations.