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Chapter 7: Problem 18

For exercises \(5-48\), simplify. $$ \frac{u^{2}}{u^{2}+6 u+8}-\frac{4}{u^{2}+6 u+8} $$

Short Answer

Expert verified
\( \frac{u-2}{u+4} \)

Step by step solution

01

Identify Common Denominator

Notice that both terms have the same denominator, which is the quadratic expression \( u^2 + 6u + 8 \). This will allow us to combine the fractions easily.
02

Combine the Numerators

Since the denominators are the same, subtract the numerators: \( \frac{u^2}{u^2+6u+8} - \frac{4}{u^2+6u+8} = \frac{u^2 - 4}{u^2 + 6u + 8} \).
03

Factor the Numerator and Denominator

Factor the numerator \( u^2 - 4 \) as \( (u+2)(u-2) \). Factor the denominator \( u^2 + 6u + 8 \) as \( (u+2)(u+4) \). Thus, the expression becomes \( \frac{(u-2)(u+2)}{(u+2)(u+4)} \).
04

Cancel the Common Factor

Cancel the common factor \( (u+2) \) from the numerator and denominator, which simplifies the expression to \( \frac{u-2}{u+4} \).

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Key Concepts

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

common denominator
When simplifying algebraic fractions, identifying a common denominator is crucial. A common denominator is the same denominator shared by all fractions in the expression.
In this exercise, both fractions already share the same denominator, which is: \(u^2 + 6u + 8\).
This simplifies our work because we don't need to modify the fractions to have a common base.
Just proceed to combine the numerators over this common denominator.
factorization
Factorization is breaking down a complex expression into simpler multiplicative components. It often helps in simplifying algebraic expressions.
In our problem, we need to factor both the numerator and the denominator to simplify the fraction further.
First, we look at the numerator: \(u^2 - 4\). This can be factored as \( (u - 2)(u + 2) \).
Next, we factor the denominator: \(u^2 + 6u + 8\), which factors to \( (u + 2)(u + 4) \).
Having factored both numerator and denominator, our fraction is now \( \frac{(u - 2)(u + 2)}{(u + 2)(u + 4)} \).
fraction simplification
Simplifying fractions involves reducing them to their simplest form.
After factoring both the numerator and the denominator in the expression, we look for any common factors to cancel out.
From our problem, we can see that both the numerator and denominator share the common factor: \(u + 2\).
By canceling the \( u + 2 \) factor from both the numerator and the denominator, the fraction becomes simpler.
canceling factors
Canceling factors is a powerful tool in simplifying algebraic fractions. It involves removing common factors from both the numerator and the denominator.
From our factored fraction \( \frac{(u - 2)(u + 2)}{(u + 2)(u + 4)} \), we see \(u + 2\) is a common factor.
Cancel this common factor to reduce the fraction to: \( \frac{u - 2}{u + 4} \).
Now, the fraction is fully simplified. Always ensure factors you cancel are common to both numerator and denominator.

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Most popular questions from this chapter

Find the number of 80,500 children and adolescents who are obese. Round to the nearest hundred. Approximately one in six children and adolescents are obese. (Source: www.cdc.gov, Nov. 2011)

The relationship of the diameter of a circle, \(x\), and the circumference of the circle, \(y\), is a direct variation. The diameter of a circle is \(20 \mathrm{~cm}\), and the circumference is \(62.8 \mathrm{~cm}\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this relationship. c. Find the circumference of a circle with a diameter of \(40 \mathrm{~cm}\).

For exercises \(67-82\), use the five steps and a proportion. In \(2010,3.5\) per 100,000 full-time equivalent workers were killed on the job with a total of 547 workers killed on the job. Find the number of full-time equivalent workers used to create this ratio. Round to the nearest whole number. (Source: www.osha.gov)

In radiography, a grid reduces the effect of X-ray scattering. The relationship of the interspace distance on the grid, \(x\), and the grid ratio, \(y\), is an inverse variation. When the interspace distance on a grid is 300 micrometers, the grid ratio is 8 . Write an equation that represents this variation. Include the units.

For exercises 87-90, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Solve: \(\frac{11}{x}+\frac{13}{12}=1\) Incorrect Answer: Least common denominator is \(12 x\). $$ \begin{aligned} 12 x\left(\frac{11}{x}+\frac{13}{12}\right) &=1 \\ 12 x\left(\frac{11}{x}\right)+12 x\left(\frac{13}{12}\right) &=1 \\ 132+13 x &=1 \\ \frac{-132}{0+13 x} &=-131 \\ \frac{13 x}{13} &=\frac{-131}{13} \\ x &=-\frac{131}{13} \end{aligned} $$
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