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Chapter 0: Problem 111

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

Short Answer

Expert verified
The student's approach is incorrect because a term can only be cancelled out from a fraction when it is a factor for all terms of the numerator and denominator in that fraction. In this case, \(x+3\) is not a factor in all terms of the fractions, so it cannot be cancelled out through division.

Step by step solution

01

Understanding the operation

Comparing both fractions, it's clear that the term \(x+3\) is present in all sequences. However, it does not appear as a common factor in any single fraction, it's part of the numerator in the first fraction and part of the denominator in the second fraction. Therefore, it cannot be cancelled out through division.
02

Realizing the incorrect operation

The student's approach of dividing both the numerator and denominator of a fraction by a common factor can be applied only when this factor is present in all terms of the numerator and the denominator. In this case, the term \(x+3\) is not common to all terms and hence, cannot be cancelled in the same division step.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Division of Rational Expressions
When dividing rational expressions, it's important to understand that these expressions are fractions where the numerator and the denominator are both polynomials. Just like with numerical fractions, if you want to divide one rational expression by another, you can multiply by the reciprocal of the expression you're dividing by.
  • To apply this, invert the divisor: flip the numerator and the denominator of the fraction you are dividing by.
  • Multiply the first rational expression by this reciprocal.
While mathematical division itself may bring to mind simple operations, division of rational expressions requires careful treatment of each term within the numerators and denominators. Always ensure that any common factors present in both the numerator and denominator are addressed correctly.
Common Factors in Algebra
In algebra, recognizing and simplifying common factors is a fundamental skill. A common factor in a polynomial is a term that divides each part of the polynomial without leaving a remainder.
  • Finding common factors helps simplify expressions, making them easier to work with.
  • If the common factor is present throughout all terms of the numerator and denominator, it can be canceled.
However, it鈥檚 crucial to understand that common factors must appear in all parts of the expression you're simplifying. For example, to divide by a factor like \(x+3\), it should be present in every term you're adjusting or cancelling, whether in the numerator or the denominator.
Simplifying Algebraic Fractions
Simplifying algebraic fractions involves reducing fractions to their simplest form. This process is similar to simplifying numerical fractions, but requires working with algebraic terms.
  • Use the greatest common factor to reduce the fraction if possible.
Remember, when simplifying algebraic fractions, always factor fully first and then cancel common factors. Incorrectly identifying or cancelling factors, as in the original exercise's approach, can result in incorrect simplification. Ensure all terms in the numerator are checked for commonality with the denominator before attempting any cancellation.

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Most popular questions from this chapter

This will help you prepare for the material covered in the next section. If 6 is substituted for \(x\) in the equation $$2(x-3)-17=13-3(x+2)$$ is the resulting statement true or false?

Each group member should research one situation that provides two different pricing options. These can involve areas such as public transportation options (with or without discount passes), cellphone plans, long-distance telephone plans, or anything of interest. Be sure to bring in all the details for each option. At a second group meeting, select the two pricing situations that are most interesting and relevant. Using each situation, write a word problem about selecting the better of the two options. The word problem should be one that can be solved using a linear inequality. The group should turn in the two problems and their solutions.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A truck can be rented from Basic Rental for \(\$ 50\) per day plus \(\$ 0.20\) per mile. Continental charges \(\$ 20\) per day plus \(\$ 0.50\) per mile to rent the same truck. How many miles must be driven in a day to make the rental cost for Basic Rental a better deal than Continental's?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In an inequality such as \(5 x+4<8 x-5,\) I can avoid division by a negative number depending on which side I collect the variable terms and on which side I collect the constant terms.

What is a compound inequality and how is it solved?
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