Problem 98 - Intermediate Algebra

Question

Write some comments to the student who wrote the following solution, explaining the error. $$ \begin{aligned} \frac{x^{2}+x-2}{x^{2}-4} \cdot \frac{x-2}{x-1} &=\frac{(x+2)(x-1)(x-2)}{(x+2)(x-2)(x-1)} \\ &=0 \end{aligned} $$

Step-by-Step Solution

Answer

The student's error was assuming the expression simplifies to zero; it actually simplifies to one.

Step 1: Analyze the Given Problem

The task involves simplifying the expression \( \frac{x^{2}+x-2}{x^{2}-4} \cdot \frac{x-2}{x-1} \). This requires factoring the numerators and denominators and then simplifying the resulting fractions by canceling common factors.

Step 2: Factor Each Expression

First, factor each part of the expression: \( x^2+x-2 = (x+2)(x-1) \) and \( x^2-4 = (x+2)(x-2) \). The second fraction, \( \frac{x-2}{x-1} \), is already in factored form.

Step 3: Set Up the Simplified Expression

Replace the numerators and denominators with their factored forms: \[ \frac{(x+2)(x-1)}{(x+2)(x-2)} \cdot \frac{(x-2)}{(x-1)}. \]

Step 4: Simplify by Canceling Common Factors

Cancel the common factors in the numerator and denominator: \( (x+2) \), \( (x-1) \), and \( (x-2) \) are common factors in the overall expression. This leaves: \[ 1. \]

Step 5: Identify the Error

The student incorrectly simplified the entire expression down to \( 0 \). In reality, after canceling common factors, it reduces to \( 1 \), not zero. The error lies in not recognizing that all factors are proportional, resulting in \( 1 \), not \( 0 \).

Key Concepts

Error Identification

Errors are easy to make and identifying them is the first step to learning from any mistake. In this case, the student simplified the rational expression down to zero, instead of one.

This happened because of misunderstanding the concept of canceling factors. When common factors in the numerator and denominator are cancelled, the values are divided by themselves, resulting in one, not zero.

• When simplifying fractions, remember that we divide common elements out, so they become one.
• If all parts of the expression cancel, we do not end up with zero, but with one.
• Always re-check your work to ensure you're simplifying correctly.

Proper error identification helps prevent misunderstandings and sharpens your skills in handling rational expressions.

Factoring Polynomials

Factoring polynomials is a crucial skill, especially when dealing with rational expressions. It's about breaking down a polynomial into simpler components.

In our example:

• The expression \(x^2 + x - 2\) can be factored into \((x+2)(x-1)\).
• The expression \(x^2 - 4\) is a difference of squares and factors into \((x+2)(x-2)\).
• The fraction \(\frac{x-2}{x-1}\) is already in its simplest form.

This process might be challenging, but you can get better with practice.

Recognizing patterns like common factors, difference of squares, and trinomials makes it easier to factor accurately, providing a strong foundation for further simplification of expressions.

Simplification Methods

Simplifying rational expressions involves reducing them to their simplest form. It's about canceling out common factors in the numerator and the denominator.

Start by factoring both the numerator and the denominator and then cancel out any factors that appear in both. Remember:

• Factor every polynomial completely first before attempting to simplify.
• Carefully cancel only common factors. Skipping this can lead to mistakes.
• The goal is to reduce the expression to its simplest terms.
• After simplification, verify your result to ensure its correctness.

Keep in mind; once every common element is canceled out, you should be left with a rational expression in its simplest form. Always review your simplification step to confirm that nothing was wrongly omitted or inaccurately retained.