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Chapter 7: Problem 49

For exercises 49-52, simplify. $$ \frac{n^{3}}{n^{2}+n-12}-\frac{27}{n^{2}+n-12} $$

Short Answer

Expert verified
\( \frac{n^2 + 3n + 9}{n + 4} \)

Step by step solution

01

Identify Common Denominator

Notice that both fractions \[ \frac{n^{3}}{n^{2}+n-12} - \frac{27}{n^{2}+n-12} \] share a common denominator \((n^2 + n - 12)\).
02

Combine the Fractions

Combine the numerators over the common denominator: \[ \frac{n^3 - 27}{n^2 + n - 12} \]
03

Recognize Difference of Cubes

Note that \(n^3 - 27\) can be factored using the difference of cubes formula: \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\] where \(a = n\) and \(b = 3\). So, \[n^3 - 27 = (n - 3)(n^2 + 3n + 9) \]
04

Factor the Denominator

Factor the quadratic expression in the denominator: \[n^2 + n - 12 = (n + 4)(n - 3) \]
05

Simplify the Fraction

Substitute the factored forms into the fraction and cancel the common factor \((n-3)\): \[ \frac{(n - 3)(n^2 + 3n + 9)}{(n + 4)(n - 3)} = \frac{n^2 + 3n + 9}{n + 4} \]

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Key Concepts

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

common denominator
To simplify algebraic fractions that have different numerators but the same denominator, the first step is to identify the common denominator. In this case, both fractions \( \frac{n^{3}}{n^{2}+n-12} - \frac{27}{n^{2}+n-12} \) share the common denominator \( (n^2 + n - 12) \). Once we've established this, we can combine the two fractions to form one single fraction. This helps us simplify the problem significantly. Remember to always verify that the denominators are indeed the same before merging the fractions.
difference of cubes
Recognizing special patterns like the difference of cubes is pivotal in algebraic simplification. The expression \( n^3 - 27 \) can be factored using the difference of cubes formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). Here, \( a = n \) and \( b = 3 \), which means: \[ n^3 - 27 = (n - 3)(n^2 + 3n + 9) \] Recognizing and applying this formula helps break down complex expressions into simpler ones. Always look for cubic terms and seek to apply this formula where appropriate.
factoring quadratics
Factoring quadratics is another essential skill for simplifying algebraic fractions. The quadratic \( n^2 + n - 12 \) in the denominator can be factored into \( (n + 4)(n - 3) \). Factoring quadratics often requires recognizing familiar patterns or using the quadratic formula. In most cases, you'll look for two numbers that multiply to the constant term (in this case, -12) and add up to the linear coefficient (in this case, 1). Ensuring your quadratic is correctly factored makes it possible to simplify fractions by canceling out common factors.
algebraic simplification
Algebraic simplification involves combining all the steps we've discussed to simplify an expression. In this problem, after merging fractions and applying the difference of cubes and factoring quadratics, we substitute the factored forms into the fraction: \[ \frac{(n - 3)(n^2 + 3n + 9)}{(n + 4)(n - 3)} \] By canceling out the common factor \( (n-3) \), the final simplified form is: \[ \frac{n^2 + 3n + 9}{n + 4} \] Simplification often involves multiple steps and skills, including recognition of patterns, factoring, and reducing fractions by canceling common factors. Practice these steps to become proficient at simplifying complex algebraic expressions.

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Most popular questions from this chapter

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)

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=3, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=9\).

For exercises \(67-82\), use the five steps and a proportion. Find the number of adults used to create the ratio "four out of five." Four out of five adults now use the Internet. 184 million adults are online from their homes, offices, schools or other locations. (Source: www.harrisinteractive.com, Nov. 17, 2008)

When the radiation is constant, the relationship of the current in an X-ray tube, \(x\), and the exposure time, \(y\), is an inverse variation. When the current is 600 milliamp, the exposure time is \(0.2 \mathrm{~s}\). Write an equation that represents this variation. Include the units.
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