Problem 111 - Introductory Algebra for College Students

Question

Use the GRAPH or TABLE feature of a graphing utility to determine if the rational expression has been correctly simplified. If the simplification is wrong, correct it and then verify your answer using the graphing utility. $$\frac{x^{2}-x}{x}=x^{2}-1, x \neq 0$$

Step-by-Step Solution

Answer

The original rational expression \(\frac{x^{2}-x}{x}\) simplifies to \(x-1\), not \(x^{2}-1\). Graphically, the function \(x-1\) is different from \(x^{2}-1\), which indicates that the initial simplification was incorrect.

Step 1: Simplify the Rational Expression

Begin by simplifying the given rational expression. After distributing the denominator \(x\) into the numerator \((x^{2}-x)\), the given rational expression simplifies to \(x-1\), not \(x^{2}-1\) as stated in the exercise.

Step 2: Graph the Rational Expression

Graph the simplified rational expression \(x-1\), along with \(x^{2}-1\) to compare differences. Remember to omit the point where \(x=0\) as stated in the exercise.

Step 3: Analysis of the Graphs

Through visual comparison, one will see that the graph of \(x-1\) does not coincide with the graph of \(x^{2}-1\). This verifies that the initial simplification provided in the exercise is incorrect. The correct simplification should have resulted in \(x-1\).

Key Concepts

Graphing Utility

A graphing utility is a powerful tool that expands the capabilities of traditional pencil-and-paper algebra. It assists in visualizing mathematical concepts, particularly rational expressions, by creating a visual representation of equations and inequalities on a coordinate plane.

When working with rational expressions, using a graphing utility helps students to compare the original and simplified forms. For instance, by plotting the rational expression \(\frac{x^2 - x}{x}\) and its purported simplified form \(x^2 - 1\), clear visual discrepancies help in identifying errors. This instant visual feedback confirms whether the simplified expression accurately represents the original equation.

Applying a graphing utility correctly also means paying attention to the conditions of the expression, like excluding points where the denominator would be zero \(x eq 0\). Overall, graphing utilities serve as an essential check mechanism and learning aid in algebra.

Rational Expression

At its core, a rational expression is a fraction wherein both the numerator and the denominator are polynomials. The word rational comes from the ratio, which signifies the expression represents a division of two quantities.

Simplifying these expressions is a fundamental aspect of algebra, and it involves reducing the expression to its lowest terms. It's akin to simplifying a fraction like \(\frac{8}{16}\) to \(\frac{1}{2}\). Similarly, a rational expression like \(\frac{x^2 - x}{x}\) can be simplified by factoring out an \(x\) from the numerator, which then gets cancelled with the \(x\) in the denominator, leaving \(x - 1\) as the simplified form.

Remember, when working with these expressions, it's essential to note the non-permissible values for \(x\), as these are values that would make the denominator zero, leading to an undefined expression.

Algebraic Simplification

The process of algebraic simplification involves rewriting expressions in a simpler or more efficient form, without changing their value. Keeping the balance between the equation's complexity and clarity is key.

In the context of rational expressions, simplification often requires factoring polynomials, canceling common factors, and recognizing identities. For the expression \(\frac{x^2 - x}{x}\), the correct simplification process includes factoring \(x\) out of the numerator and cancelling it out against the \(x\) in the denominator. The correct simplification would therefore be \(x - 1\), not \(x^2 - 1\).

Common errors in simplification can be caused by over-cancellation or misapplying the distributive property. Constant practice, alongside verification tools like graphing utilities, helps in mastering this important algebraic skill.