Problem 62 - Elementary Algebra

Question

Which of the following simplification processes are correct? Explain your answer. $$ \frac{2 x}{x}=2 \quad \frac{x+2}{x}=2 \quad \frac{x(x+2)}{x}=x+2 $$

Step-by-Step Solution

Answer

Expression 1 and 3 are correct. Expression 2 is incorrect.

Step 1: Simplify Each Expression Separately

We'll start by simplifying each expression one at a time to determine if the simplifications are correct.1. **Expression 1**: \( \frac{2x}{x} \) - The terms \( x \) in the numerator and denominator can cancel out (assuming \( x eq 0 \)). - This simplifies to \( 2 \), which matches the given simplification. This process is correct.2. **Expression 2**: \( \frac{x+2}{x} \) - This expression cannot be simplified to \( 2 \) because there is no common factor of \( x \) in both the numerator and the terms are not separated by multiplication. The correct simplification should remain \( \frac{x+2}{x} \). The simplification provided is incorrect.3. **Expression 3**: \( \frac{x(x+2)}{x} \) - Here, \( x \) in the numerator and \( x \) in the denominator can be canceled out (assuming \( x eq 0 \)). - This simplifies to \( x+2 \), which matches the given simplification. This process is correct.

Step 2: Identify the algebraic structure

Determine the type of algebraic problem.

Step 3: Apply algebraic techniques

Use factoring, expanding, or systematic methods.

Step 4: Simplify and solve

Simplify expressions and solve for unknowns.

Step 5: State the result

Write the final answer.

Step 6: Conclude with the answer

Expression 1 and 3 are correct. Expression 2 is incorrect.

Key Concepts

Simplification

Simplification in algebra refers to making an expression easier to read and understand by reducing it to its simplest form. This involves combining like terms, performing the operations, and eliminating unnecessary components within the expression.

A simplified expression is more straightforward and often requires fewer steps to work with in equations or further calculations.

• For instance, in the expression \( \frac{2x}{x} \), the simplification process involves recognizing that the \( x \) in the numerator and the denominator can be canceled out, resulting in a more manageable expression, \( 2 \).

Always remember to ensure the expressions you work with can actually be simplified based on mathematical rules, as some expressions like \( \frac{x+2}{x} \) do not have the same common factors to allow further simplification without changing their value.

Canceling Terms

Canceling terms is an integral step in simplifying algebraic expressions. It involves removing identical terms found in both the numerator and the denominator of a fraction to reduce the expression to its simplest form.

This technique is valid only when the term to be canceled is a factor of both the numerator and the denominator. It is essential to ensure the canceled terms do not equate to zero since division by zero is undefined.

• In the expression \( \frac{x(x+2)}{x} \), each \( x \) from the numerator and denominator can be canceled because \( x \) is a common factor (provided \( x eq 0 \)).
• However, \( \frac{x+2}{x} \) cannot cancel \( x \) because it is not a standalone factor of the numerator terms and not all terms are multiplied by \( x \).

When attempted correctly, canceling terms is a powerful method to simplify fractions in algebra.

Factors

When dealing with factors in algebra, understanding how parts of an expression can be multiplied together to produce the entire expression is essential. Factors are the building blocks of an expression, involving the numerical and variable elements that combine to form it.

To simplify or factor an expression, identify common factors in the terms. In rational expressions, knowing the factors will help in simplifying or canceling them out where applicable.

• Consider the expression \( \frac{2x}{x} \); the factors are \( 2 \) and \( x \), and since \( x \) is present in both parts of the fraction, it can be factored and canceled.

Recognizing and working with factors is crucial in solving algebraic expressions, equations, and especially in simplifying rational expressions.

Rational Expressions

A rational expression is essentially a ratio of two polynomials, similar in format to a fraction, where both the numerator and the denominator consist of polynomial expressions. The key operations with rational expressions involve manipulation by adding, subtracting, multiplying, or dividing.

When simplifying, understanding the relationship between the numerator and denominator is crucial. They might require factoring first or finding common terms to simplify further.

• In our example, expressions like \( \frac{x(x+2)}{x} \) are rational because they involve the ratio of polynomial expressions. The polynomial \( x(x+2) \) in the numerator can be divided by the polynomial \( x \) in the denominator to reduce the expression.

Always be alert for restrictions such as not dividing by zero, as this can change the domain and validity of the expression. Rational expressions offer a wide scope for simplification and are vital in much of algebra.