Problem 1 - Essential Calculus

Question

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{5}{x^{2}-10 x} $$

Step-by-Step Solution

Answer

The partial fraction form of the given expression is \(\frac{A}{x} + \frac{B}{x - 10}\).

Step 1: Factorize the denominator

First of all, simplify the expression by factorizing \(x^{2}-10x\) in the denominator. This can be written as \(x \cdot (x - 10)\).

Step 2: Write the partial fraction form

Next, express the fraction in partial fraction form with unknown constants. This means, the given rational expression can be written as \(\frac{A}{x} + \frac{B}{x - 10}\), where A and B are constants to determine.

Step 3: Resulting form

Hence, the form of the partial fraction decomposition of the rational expression is \(\frac{A}{x} + \frac{B}{x - 10}\). We are asked not to solve for the constants A and B in this exercise.

Key Concepts

Rational Expression

A rational expression is like a fraction but with polynomials. Think of it as a fraction where the numerator and denominator are polynomial expressions. The word "rational" comes from "ratio," indicating a relationship between two quantities. Understanding rational expressions is crucial in algebra because they appear frequently in mathematical scenarios. Just like regular fractions, we can add, subtract, multiply, and divide them—keeping in mind the need for a common denominator during addition and subtraction processes.
To work with rational expressions efficiently, it's important to simplify them first by factoring. This simplification can provide us with some advantages, such as easier computation and the revelation of simpler equivalents. Once simplified, you can perform operations or apply transformations such as partial fraction decomposition, which helps in breaking down complex expressions into more manageable parts.

Factorization

Factorization is the process of breaking down a complex expression into simpler components that, when multiplied together, yield the original expression. It's a vital tool in algebra used to simplify expressions and solve equations. In the exercise, we factorized the polynomial in the denominator, \(x^2 - 10x\), by noting it can be rewritten as \(x \cdot (x - 10)\).
By performing factorization:

• You identify common factors that can be taken out of the expression.
• It becomes much easier to handle subtraction or division among rational expressions.
• Factorization is also a prerequisite for the partial fraction decomposition which helps in rewriting complicated expressions into simpler parts.

Factorizing helps us understand the roots of the polynomial when we set each factor equal to zero. In our example, solving \(x(x-10) = 0\) gives the roots at \(x = 0\) and \(x = 10\), indicating important points where the expressions behave differently.

Denominator

The denominator is the bottom part of a fraction or rational expression. In a rational expression like the one given in our exercise, \(\frac{5}{x^2 - 10x}\), the denominator decides the values for which the expression is undefined. Critical for understanding the domain of the function, the denominator cannot be zero.In our example, the expression \(x^2 - 10x\) as the denominator requires careful examination:

• For simplicity and problem-solving ease, we factorized it to \(x \cdot (x - 10)\).
• This factorized form reveals the values \(x = 0\) and \(x = 10\) where the expression becomes undefined.
• The denominator essentially sets up the conditions for any solutions related to the expression, restrictions, or simplifying methods such as partial fraction decomposition.

When working with fractions or rational expressions, always pay attention to the denominator, as it influences the solution strategy and tells us about the domains or areas of exclusion in a function.