Problem 100 - College Algebra

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 both numerator and denominator by \(x+3\) would invalidate the given expressions.

Step 1: Identify the Two Fractions/Expressions

The problem presents two expressions: the first is \(\frac{7 x}{x+3}\) and the second is \(\frac{(x+3)^{2}}{x-5}\). Doing the division operation means to find the quotient when the first expression is divided by the second expression.

Step 2: Apply the Division Rule for Fractions

Use the rule of division: division between fractions is equivalent to multiplying the dividend by the reciprocal of the divisor. Thus, \(\frac{7 x}{x+3} \div \frac{(x+3)^{2}}{x-5}\) becomes \(\frac{7 x}{x+3} \times \frac{x-5}{(x+3)^{2}}\).

Step 3: Examine the Claim About Common Factors

Upon close examination, it's clear that both the simplified division expression and original division expression don't allow for \(x+3\) to be cancelled out in both numerator and denominator because in the simplified version \(x+3\) is squared in the denominator. Division is only possible when the expression is present in both numerator and denominator in the same power.

Step 4: Final Review of the Statement

Based on the performed simplification, the claim in the question, 'divide the numerator and denominator by the common factor \(x+3\)' is wrong because dividing by the common factor would result in a false expression.

Key Concepts

Division of Polynomials

Understanding how to divide polynomials is a fundamental skill in algebra, particularly when dealing with complex expressions. The division of polynomials can be compared to the long division we learn with numbers, but instead of numbers, we are working with variables raised to different powers.

Let's take the given exercise as an example. You have \(\frac{7 x}{x+3} \div \frac{(x+3)^{2}}{x-5}\). Here, the operation might look daunting due to the presence of polynomials in both the numerator and the denominator. However, if we apply the division rule correctly, we turn a division problem into a multiplication problem by using the reciprocal of the second fraction.

Key Steps in Polynomial Division

• Identify the dividend and the divisor. In our case, \(\frac{7 x}{x+3}\) is the dividend and \(\frac{(x+3)^{2}}{x-5}\) is the divisor.
• Find the reciprocal of the divisor to convert the division into multiplication. The reciprocal of \(\frac{(x+3)^{2}}{x-5}\) is \(\frac{x-5}{(x+3)^{2}}\).
• Multiply the dividend by the newly found reciprocal of the divisor.
• Simplify the resulting expression, if possible.

Performing these steps properly is crucial for a correct solution. In our initial exercise, the idea of dividing by the common factor \(x+3\) was incorrect because the divisor featured \(x+3\) squared, which changes the terms significantly.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is much like cleaning a cluttered room: the goal is to make the expression as neat and tidy as possible. This involves combining like terms, reducing fractions, and canceling out common factors where applicable.

However, simplification must follow the rules of algebra meticulously. Key to this is recognizing that only like terms—that is, terms with the same variable to the same power—can be combined, and common factors can only be canceled when they appear in both the numerator and the denominator of a fraction.

Common Mistakes in Simplification

While simplifying, it's easy to jump to conclusions too quickly. For instance, some might think that \(x+3\) in \(\frac{7 x}{x+3}\) can be canceled with \(x+3\) in \(\frac{(x+3)^{2}}{x-5}\). However, that's not the case since in the divisor, \(x+3\) is squared, making it a different term. Always ensure that when canceling terms, the expressions are identical. If they are not, you may end up with a completely different and incorrect expression.

Reciprocal of a Fraction

The reciprocal of a fraction is simply flipping the numerator and the denominator. If you have a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). Knowing the reciprocal is particularly useful when dividing fractions, as you replace division with multiplication by the reciprocal of the divisor.

The concept of reciprocals in algebra takes us back to one of the fundamental rules of arithmetic: dividing by a number is the same as multiplying by its reciprocal. This principle allows us to transform division problems into multiplication ones, which are often simpler to execute.

For the reciprocal of algebraic fractions, care must be taken when the expressions involve powers. For example, in our exercise's divisor \(\frac{(x+3)^{2}}{x-5}\), the reciprocal is not \(\frac{x-5}{x+3}\) but rather \(\frac{x-5}{(x+3)^{2}}\) because the entire expression \(x+3\) is squared. Misidentifying the reciprocal can lead to incorrect answers and further confusion in the division process.