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Chapter 7: Problem 86

Explain why the LCD of the rational expressions \(\frac{7}{x+1}\) and \(\frac{9 x}{(x+1)^{2}}\) is \((x+1)^{2}\) and \(\operatorname{not}(x+1)^{3}\).

Short Answer

Expert verified
The LCD is \((x+1)^{2}\) as it's the highest power of \(x+1\) in both denominators.

Step by step solution

01

Identify the Denominators

The denominators of the given rational expressions are \(x+1\) for \(\frac{7}{x+1}\) and \((x+1)^{2}\) for \(\frac{9x}{(x+1)^{2}}\).
02

Find the Least Common Denominator (LCD)

The LCD is the smallest expression that is a multiple of all the denominators. Since one denominator is \(x+1\) and the other is \((x+1)^{2}\), the LCD must include the highest power of \(x+1\) found in these denominators.
03

Determine the Highest Power of the Common Factor

The factors in the denominators are \(x+1\) and \((x+1)^2\). The highest power of \(x+1\) in these expressions is \((x+1)^2\) found in the expression \(\frac{9x}{(x+1)^{2}}\).
04

Explain Why the LCD is Not \\(x+1\\)^3

The expression \((x+1)^3\) would be considered if one denominator included a cube power of \(x+1\), but since the highest power present is only \(2\), \(x+1)^{2}\) correctly represents the LCD. Using \((x+1)^{3}\) would introduce unnecessary factors, making it no longer the 'least' common denominator.

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Key Concepts

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

Rational Expressions
Rational expressions are fractions that have polynomials in both their numerators and denominators. In this context, we're discussing two rational expressions:
  • One is \(\frac{7}{x+1}\), which has a simple polynomial in the denominator.
  • The other is \(\frac{9x}{(x+1)^{2}}\), where the denominator is a squared polynomial.
Understanding these expressions is crucial, especially when performing operations like addition or subtraction. Before proceeding, denominators of the given expressions must be the same. Hence, finding a common denominator is a key step in working with rational expressions.
Denominators
In the world of fractions and rational expressions, denominators play a critical role. They are the bottom part of the fraction, dictating the division of the numerator. In the given example, we have two denominators:
  • \(x+1\) from the first expression \(\frac{7}{x+1}\)
  • \((x+1)^2\) from the second expression \(\frac{9x}{(x+1)^{2}}\)
Finding a least common denominator (LCD) involves determining the smallest possible denominator that can accommodate both these terms. When handling rational expressions, aligning denominators by using the LCD enables methods like addition and subtraction to remain valid.
Highest Power
To find the least common denominator, identifying the highest power of each polynomial factor in the given denominators is necessary. Let's break it down:
  • The denominator \(x+1\) has a power of 1 in the expression \(\frac{7}{x+1}\).
  • The denominator \((x+1)^2\) has a power of 2 in the expression \(\frac{9x}{(x+1)^2}\).
The highest power is simply the largest exponent of the polynomial appearing in any of the denominators. Therefore, the highest power of \(x+1\) seen here is 2, making \((x+1)^2\) our candidate for the least common denominator.
Factoring
Factoring is a fundamental process in algebra, especially when simplifying expressions or finding the least common denominator. The essence of factoring lies in breaking down polynomials into their simplest forms or structures:
  • \(x+1\) is a factor of the first expression's denominator.
  • \((x+1)^2\) incorporates \(x+1\) as a repeated factor.
Once we factorize each denominator involved, we can easily spot the common factors and their respective powers. The highest power of these common factors is then chosen to construct the least common denominator.
Multiples
When discussing multiples in the context of rational expressions, we're focusing on how each denominator can be expanded to form common multiples. A multiple of a term is essentially that term raised to, or multiplied by, any integer power greater than zero:
  • \((x+1)\) can be considered as a multiple trivially multiplied by itself (i.e., \((x+1)^1\)).
  • \((x+1)^2\) is a further multiple of \((x+1)\).
The "least" common denominator results from selecting the smallest expression that is a multiple of all denominators involved. It ensures the simplest form that can handle all factors present, without unnecessary expansion like \((x+1)^3\), which adds complexity without benefiting the expression鈥檚 equivalence.

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Most popular questions from this chapter

Solve. See the Concept Check in the Section. Which of the following are equivalent to \(\frac{\frac{a}{7}}{\frac{b}{13}} ?\) a. \(\frac{a}{7} \cdot \frac{b}{13}\) b. \(\frac{a}{7} \div \frac{b}{13}\) c. \(\frac{a}{7} \div \frac{13}{b}\) d. \(\frac{a}{7} \cdot \frac{13}{b}\)

One of the great algebraists of ancient times a man named Diophantus. Litle is known of his life other than the lived and worked in Alexandria. Some historians believe he lived during the first century of the Christian era, about the time of Nero. The only che to his personal life is the following epigram found in a collection called the Palatine A nthology. God granted him youth for a sixth of his life and added a twelfth part to this. He clothed his cheeks in down. He lit him the light of wedlock after a seventh part and five years after his marriage. He granted him a son. Alas, lateborn wretched child. After attaining the measure of half his father's life, cruel fate overtook him, thus leaving Diophantus during the last four years of his life only such consolation as the science of numbers. How old was Diophantus at his death?" We are looking for Diophantus' age when he died, so let x represent that age. If we sum the parts of his life, we should get the total age. Parts of his life \(\left\\{\begin{array}{l}{\frac{1}{6} x+\frac{1}{12} x \text { is the time of his youth. }} \\ {\frac{1}{7} x \text { is the time between his youth and when }} \\ {\text { he married. }} \\ {5 \text { years is the time between his marriage }} \\ {\text { and the birth of his son. }} \\\ {\frac{1}{2} x \text { is the time Diophantus had with his son. }} \\ {4 \text { years is the time between his son's death }} \\ {\text { and his own. }}\end{array}\right.\) The sum of these parts should equal Diophantus' age when he died. $$ \frac{1}{6} \cdot x+\frac{1}{12} \cdot x+\frac{1}{7} \cdot x+5+\frac{1}{2} \cdot x+4=x $$ How old was Diophantus when his son was born? How old was the son when he died?

Perform each indicated operation. Explain when the LCD of the rational expressions in a sum is the product of the denominators.

Perform the indicated operations. Addition, subtraction, multiplication, and division of rational expressions are included here. $$ \frac{15 x}{x+8} \cdot \frac{2 x+16}{3 x} $$

Solve the following. See Section 3.6. If \(f(x)=x^{2}-6,\) find \(f(-1)\).
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