Problem 4 - Precalculus: Mathematics for Calculus

Question

Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$\frac{x}{x^{2}+3 x-4}$$

Step-by-Step Solution

Answer

\(\frac{A}{x-1} + \frac{B}{x+4}\)

Step 1: Factor the Denominator

First, identify if the denominator is factorable into simpler linear or quadratic factors. The denominator is \(x^2 + 3x - 4\). We need to find factors of \(-4\) that add up to \(3\). The factors are \(4\) and \(-1\). Hence, the denominator factors into \((x - 1)(x + 4)\).

Step 2: Set Up the Partial Fraction Decomposition

Since \((x - 1)\) and \((x + 4)\) are distinct linear factors, the partial fraction decomposition is set up as follows: \[\frac{x}{x^{2}+3x-4} = \frac{A}{x-1} + \frac{B}{x+4}\] where \(A\) and \(B\) are constants to be determined.

Key Concepts

Factoring Quadratics

Factoring quadratics is a crucial step in partial fraction decomposition, especially when dealing with rational expressions. In our original exercise, we are given a quadratic expression in the denominator: \(x^2 + 3x - 4\). The process of factoring involves finding two numbers that multiply to give the constant term, \(-4\), and add up to the linear coefficient, \(3\). In this case, \(4\) and \(-1\) are the numbers that meet these criteria. This allows us to rewrite the expression as \((x - 1)(x + 4)\).

Simple strategies for factoring quadratics include:

• Identifying pairs of factors of the constant term.
• Checking which pair sums to the linear coefficient.
• Rewriting the quadratic as a product of two binomials.

After factoring, the decomposition of the expression becomes much more straightforward because we can now separate the expression into simpler fractions. Recognizing factorable forms quickly can save time and simplify many mathematical problems.

Linear Factors

Linear factors are expressions of the form \(ax + b\). These are essentially lines when plotted on a graph, and they simplify the process of breaking down rational expressions. In the case of our problem, the linear factors that resulted from factoring the quadratic \(x^2 + 3x - 4\) are \((x - 1)\) and \((x + 4)\).

Key points about linear factors include:

• Every linear factor represents a root (or zero) of the equation.
• These factors are used to set up partial fractions.
• The coefficients in front of the linear expressions adjust the steepness of the line.

When setting up partial fraction decomposition, each linear factor forms the denominator of its own fraction. For the decomposition of \(\frac{x}{x^2 + 3x - 4}\), the expression is broken into fractions \(\frac{A}{x-1}\) and \(\frac{B}{x+4}\), showcasing the usefulness of linear factors.

Rational Expressions

Rational expressions are ratios of polynomials. They resemble fractions in that they involve a numerator and a denominator. In our context, the given expression \(\frac{x}{x^2 + 3x - 4}\) is a rational expression. Understanding them is vital because they can be simplified under certain conditions by canceling common factors, or decomposed into simpler parts using partial fraction decomposition, as demonstrated in our exercise.

Essential knowledge about rational expressions:

• They include polynomials of any degree in both numerator and denominator.
• Simplifying them might involve factoring, like we did with the denominator here.
• The goal can be to make them easier to integrate or differentiate.

By decomposing a complex rational expression into simpler components, we are often able to work through calculations with more ease. In calculus, this step aids in the integration process, which is one of the main applications of partial fraction decomposition.