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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

A group of friends agrees to share the cost of a \(\$ 480,000\) vacation condominium equally. Before the purchase is made, four more people join the group and enter the agreement. As a result, each person's share is reduced by \(\$ 32,000 .\) How many people were in the original group?

The bar graph shows median yearly earnings of full-time workers in the United States for people 25 years and over with a college education, by final degree earned. Exercises \(3-4\) are based on the data displayed by the graph. (graph can't copy) The median yearly salary of an American whose final degree is a master's is \(\$ 70\) thousand less than twice that of an American whose final degree is a bachelor's. Combined, two people with each of these educational attainments earn \(\$ 173\) thousand. Find the median yearly salary of Americans with each of these final degrees.

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$8 x-2 \geq 14$$

You had \(\$ 10,000\) to invest. You put \(x\) dollars in a safe, government- insured certificate of deposit paying \(5 \%\) per year. You invested the remainder of the money in noninsured corporate bonds paying \(12 \%\) per year. Your total interest earned at the end of the year is given by the algebraic expression $$ 0.05 x+0.12(10,000-x) $$ a. Simplify the algebraic expression. b. Use each form of the algebraic expression to determine your total interest earned at the end of the year if you invested \(\$ 6000\) in the safe, government-insured certificate of deposit.

A new car worth \(\$ 24,000\) is depreciating in value by \(\$ 3000\) per year. a. Write a formula that models the car's value, \(y,\) in dollars, after \(x\) years. b. Use the formula from part (a) to determine after how many years the car's value will be \(\$ 9000\).
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