Problem 100 - Algebra and Trigonometry

Question

determine whether each statement makes sense or does not make sense, and explain your reasoning. When performing the division $$ \frac{7 x}{x+3} \div \frac{(x+3)^{2}}{x-5} $$ I began by dividing the numerator and the denominator by the common factor, \(x+3\)

Step-by-Step Solution

Answer

The statement does not make sense because dividing the numerator and the denominator by the common factor (\(x+3\)) is not the correct operation to simplify the division of two fractions. The correct process is to multiply by the reciprocal, not divide by a common factor.

Step 1: Identify the Given Fractions

Identify the two fractions provided, which are \(\frac{7x}{x+3}\) and \(\frac{(x+3)^2}{x-5}\). The operation involved is division.

Step 2: Verify the Procedure

Check if the procedure mentioned, i.e., dividing the numerator and the denominator by the common factor (\(x+3\)), can be applied. Note that the numerator of the first fraction and the denominator of the second fraction can be divided by (\(x+3\)).

Step 3: Implement the Procedure

Implement the procedure as given by: \(\frac{7x}{x+3} \div \frac{(x+3)^2}{x-5} = \frac{\frac{7x}{x+3}}{\frac{(x+3)^2}{x-5}} = \frac{7x}{x+3} \cdot \frac{x-5}{(x+3)^2}\). We simplify to get \(\frac{7x(x-5)}{(x+3)(x+3)}\).

Step 4: Conclusion

The original statement does not make sense because the common factor (\(x+3\)) was not correctly removed from the entire expression. Instead, both fractions were multiplied resulting in a different outcome compared to dividing both the numerator and the denominator by the common factor.

Key Concepts

Algebraic Fractions

Algebraic fractions are similar to regular fractions, but they include variables like 'x' or 'y' in their numerators, denominators, or both. Just as with numerical fractions, the main principle of working with algebraic fractions is to find a common factor that can be used to simplify them.

When dealing with algebraic fractions such as \(\frac{7x}{x+3}\), it's important to look for opportunities to simplify the fraction by canceling out any common factors in the numerator and denominator. This process is particularly useful when multiplying or dividing fractions. To divide algebraic fractions, you invert the second fraction and then proceed to multiply. However, before you multiply, always look to reduce the fractions to their simplest form by canceling out any common terms. This step is crucial for avoiding more complex calculations and making the problem easier to solve.

Complex Rational Expressions

Complex rational expressions are fractions that contain a fraction in either the numerator, denominator, or both. These expressions can sometimes seem daunting to simplify due to their layered structure, but they follow the same fundamental rules as simpler fractions.

For example, when dividing complex rational expressions such as \(\frac{7x}{x+3} \) by \(\frac{(x+3)^2}{x-5}\), the key step is to multiply the first expression by the reciprocal of the second. Consequently, you get a single complex rational expression, which then can be further simplified. To simplify, you look for and cancel out common terms in the numerator and denominator—being particularly careful not to cancel out terms that should be kept. Simplifying complex rational expressions often involves factoring polynomials, recognizing common terms, and being meticulous in arithmetic operations.

Simplifying Expressions

Simplifying expressions in algebra is the process of reducing them to their most basic form without changing the value of the original expression. This includes combining like terms, factoring, cancelling common factors, and performing arithmetic operations efficiently and correctly.

When simplifying the division of algebraic fractions such as \(\frac{7x(x-5)}{(x+3)(x+3)}\), the objective is to reduce the expression to the simplest terms possible. This may involve distributing terms across parentheses, combining like terms, and reducing fractions. It is vital to ensure that you do not remove components that are not common factors across the entire expression, as that would alter the value of the expression. Efficiency in simplification comes from recognizing patterns and having a strong grasp of basic algebraic principles. The process of simplification can make complex problems more approachable and reveal relationships between algebraic concepts that may not be immediately apparent.