Problem 16 - Algebra and Trigonometry

Question

write the partial fraction decomposition of each rational expression. $$\frac{x}{x^{2}+2 x-3}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the rational expression \(\frac{x}{x^{2}+2x-3}\) is \(\frac{3/4}{x+3} + \frac{1/4}{x-1}\).

Step 1: Factor the denominator

In order to facter the denominator \(x^{2}+2x-3\), we look for two numbers that add up to 2 (the coefficient of \(x\)) and multiply to -3 (the constant term). These numbers are 3 and -1. Therefore, the factorized expression is \((x+3)(x-1)\).

Step 2: Write the partial fraction decomposition

We write the expression \(\frac{x}{(x+3)(x-1)}\) as the sum of two fractions: \(\frac{A}{x+3} + \frac{B}{x-1}\). The original numerator \(x\) must then be equal to the sum of the numerators of these two fractions, ie. \(x=A(x-1)+B(x+3)\).

Step 3: Solve for the unknowns A and B

We can solve the equation \(x=A(x-1)+B(x+3)\) for \(A\) and \(B\) by substituting convenient values of \(x\) to simplify the calculations. If \(x=1\), then \(A*0 + B*4 = 1\) giving us \(B=\frac{1}{4}\). If \(x=-3\), then \(A*-4 + B*0 = -3\), so \(A=\frac{3}{4}\).

Key Concepts

Rational Expression

A rational expression is like a fraction, but instead of integers as the numerator and denominator, it has polynomials. In general, a rational expression takes the form \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials. It's essential to understand that just like in a simple fraction, the polynomial in the denominator must not be zero, as it would make the expression undefined.
Rational expressions are useful in various algebraic problems, especially when dealing with quotient relationships between polynomial functions. They can be simplified, added, subtracted, multiplied, and divided, following similar rules as those for numeric fractions. Learning to manipulate these expressions can help solve complex algebraic equations and problems.

Factoring Polynomials

Factoring polynomials is a critical skill in algebra that involves rewriting a polynomial as a product of its simplest components, called factors. This decomposition helps in simplifying expressions and solving equations more conveniently.
To factor a quadratic polynomial like \(x^2 + 2x - 3\), we need to find two numbers that both add up to the linear coefficient (in this case, 2) and multiply to the constant term (here, \(-3\)). These two numbers are 3 and \(-1\). Thus, the polynomial can be decomposed into \((x + 3)(x - 1)\).
Factoring is foundational for techniques like partial fraction decomposition, making it easier to handle complex rational expressions by breaking them into smaller, manageable parts.

Solving Equations

Solving equations is all about finding values that make the equation true. In the context of partial fraction decomposition, this often involves solving for unknown coefficients in expressions. Here, we deal with an equation arising from the decomposition, such as \(x = A(x - 1) + B(x + 3)\).
The goal is to determine the values of \(A\) and \(B\) to express the rational expression as a sum of simpler fractions like \(\frac{A}{x+3} + \frac{B}{x-1}\). To find these coefficients efficiently, one straightforward strategy is to substitute specific values of \(x\) that eliminate one of the terms, making calculations simpler.
For instance, substituting \(x = 1\) quickly gives \(B\), while \(x = -3\) yields \(A\). Such techniques streamline the process of solving for unknown coefficients in algebraic fractions.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, denominator, or both are algebraic expressions, usually polynomials. They play a significant role in algebra, especially when simplifying complex expressions or solving equations.
The process of working with algebraic fractions often requires methods to rewrite them in simpler terms. This is where partial fraction decomposition is beneficial. By expressing a complex rational expression as a sum of simpler fractions, it becomes more straightforward to integrate or differentiate, should the need arise.
In algebraic fractions, always remember the importance of the denominator not being zero and the power of factoring it as part of simplifying or decomposing fractions. Managing algebraic fractions well can lead to significant simplifications in mathematical expressions and solutions.