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Chapter 7: Problem 82

For exercises 39-82, simplify. $$ \frac{3 p^{2}+15 p+12}{p^{2}+4 p+3} \div \frac{18}{6 p+18} $$

Short Answer

Expert verified
\(p+4\).

Step by step solution

01

Factorize the Numerator of the First Fraction

Factor the numerator \(\frac{3p^2 + 15p + 12}{p^2 + 4p + 3}\). Find the roots by factoring: \(\frac{3(p^2 + 5p + 4)}{p^2 + 4p + 3} \). Further breaking it down: \(\frac{3(p+3)(p+4)}{p^2 + 4p + 3} \).
02

Factorize the Denominator of the First Fraction

Factor the denominator: \(\frac{3(p+3)(p+4)}{(p+3)(p+1)} \). The first fraction now is: \(\frac{3(p+4)}{(p+1)} \).
03

Factorize the Second Fraction

Rewrite the second fraction \(\frac{18}{6p+18}\) and factor it. Simplify the denominator: \(\frac{18}{6(p+3)}\) becomes: \(\frac{3}{p+3}\).
04

Division into Multiplication

Convert the division into multiplication: \(\frac{3(p+4)}{(p+1)} \times \frac{p+3}{3} \).
05

Simplification

Simplify the multiplication: \(\frac{(p+4)(p+3)}{(p+1)} \times \frac{1}{1} = \frac{(p+4)}{1} = (p+4)\).

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Key Concepts

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

factoring polynomials
When working with algebraic fractions, one crucial step is factoring polynomials. Factoring helps to break down complex expressions into simpler, more manageable parts. For instance, in the given exercise, we begin by factoring the numerator 3p^2 + 15p + 12. Identifying common factors or using methods like the quadratic formula can simplify this process. In this case, we find that 3(p+3)(p+4) is the factorized form. This step reveals the roots of the polynomial, making it easier to simplify the fraction.
rational expressions
A rational expression is simply a fraction where the numerator and/or the denominator are polynomials. Understanding rational expressions is essential for simplifying algebraic fractions. Just like with regular fractions, you can add, subtract, multiply, and divide rational expressions by following specific rules. In the given problem, both the initial fractions are rational expressions: \(\frac{3(p+3)(p+4)}{(p+3)(p+1)}\) and \(\frac{18}{6(p+3)}\). Simplifying rational expressions often involves factoring polynomials and reducing common factors.
multiplication of fractions
To multiply fractions is straightforward: multiply the numerators together and the denominators together. However, when dealing with algebraic fractions, first simplify both fractions if possible. In our exercise, we simplify \(\frac{3(p+4)}{(p+1)}\) and \(\frac{p+3}{3}\) before multiplication: \(\frac{3(p+4)}{(p+1)} \times \frac{p+3}{3}\). Notice how the 3 cancels out in the numerator and denominator, simplifying the expression to \(\frac{(p+4)(p+3)}{(p+1)}\). This reduces the complexity of further calculations.
division of fractions
Dividing fractions involves multiplying by the reciprocal of the divisor. This means flipping the second fraction and then multiplying. In our exercise, we convert the division into multiplication: \(\frac{3(p+4)}{(p+1)} \times \frac{p+3}{3}\). Conversion simplifies the complex operation, turning it into a problem involving the multiplication of fractions. Always remember to simplify both fractions before multiplying for the easiest computation.

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Most popular questions from this chapter

When the top of a cone is removed, the formula for the volume of the remaining cone (the frustrum) is \(V=\frac{1}{3} \pi\left(R^{2}+R r+r^{2}\right) h\), where \(r\) is the radius of the circle at the top of the frustrum and \(R\) is the radius of the circle at the bottom of the frustrum. In 1856, an American army officer, Henry Hopkins Sibley, invented and received a patent for the design of a conical tent that could sleep 12 soldiers. (The apex is the diameter of the top of the frustrum.) Find the volume of the tent in cubic feet. Use \(\pi \approx 3.14\). Round to the nearest whole number. Be it known that I, H.H. Sibley, United States Army, have invented a new and improved Conical Tent ... the tent is in shape the frustrum of a cone; the base 18 feet; the height 12 feet; the apex 1 foot 6 inches [1.5 ft]. (Source: patimg1.uspto.gov)

For exercises 59-66, use the five steps. Assume that the rate of work does not change if done individually or together. The water from a garden hose turned on at full pressure fills a hot tub in \(45 \mathrm{~min}\). If the drain is open, the hot tub empties in \(62 \mathrm{~min}\). Find the amount of time to fill the hot tub with the drain open. Round to the nearest whole number.

Identify the slope of the line represented by $$ y=\left(\frac{40 \mathrm{mi}}{1 \mathrm{hr}}\right) x $$

For exercises \(67-82\), use the five steps and a proportion. Find the number of 725,000 women in their mid \(-40 \mathrm{~s}\) with a history of normal pregnancy who would be expected to have a heart attack or stroke some 10 years later. Of 100 women in their mid-40's with a history of normal pregnancy, about 4 would be expected to have a heart attack or stroke some 10 years later. (Source: www.nytimes.com, March 17, 2009)

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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