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Chapter 7: Problem 12

Multiply as indicated. $$\frac{x^{2}-49}{x^{2}-4 x-21} \cdot \frac{x+3}{x}$$

Short Answer

Expert verified
The simplified form of the multiplication of rational expressions is \[\frac{x^{2} + 7x}{x}\] with restrictions on \(x\) being \(x鈮�0, x鈮�-3, x鈮�7\).

Step by step solution

01

Factorise the polynomials

Factorize the polynomials in the numerator and the denominator. \(x^{2}-49\) can be factorised into \((x-7)(x+7)\). Likewise, \(x^{2}-4x-21\) can be factorised and written as \((x-7)(x+3)\). After simplifying, the equation becomes \[\frac{(x-7)(x+7)}{(x-7)(x+3)} \cdot \frac{x+3}{x}\].
02

Simplify the rational expressions

Next, we simplify the equation by cancelling out the common factors in the numerators and denominators. Here, we can cancel out \((x-7)\) and \((x+3)\) from the numerator and denominator, respectively. We then get \[\frac{(x+7)}{x}.\]
03

Multiply the simplified fractions

The final step is to multiply the remaining fractions. We consider the numerator of the first fraction as the complete numerator and the denominator of the other fraction as the full denominator. The multiplication results in \[\frac{x^{2} + 7x}{x}.\]
04

Identify any restrictions on the variable

Identify the values for which x would make the denominator zero, as this would make the fraction undefined. Here, \(x\) cannot equal zero, since this would make the denominator zero. Moreover, the original rational expressions would be undefined for \(x=7\) and \(x=-3\). Therefore, the restrictions are \(x鈮�0, x鈮�-3, x鈮�7\).

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Most popular questions from this chapter

perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\text { Simplify: } \frac{(y+2) y-2 \cdot 4}{4 y(y+4)}$$

If you know how many hours it takes for you to do a job, explain how to find the fractional part of the job you can complete in \(x\) hours.

Add or subtract as indicated. Simplify the result, if possible. $$\frac{3}{x}+\frac{4}{x-6}$$

The temperature, in degrees Fahrenheit, of a dessert placed in a freezer for \(t\) hours is modeled by $$ \frac{t+30}{t^{2}+4 t+1}-\frac{t-50}{t^{2}+4 t+1} $$ a. Express the temperature as a single rational expression. b. Use your rational expression from part (a) to find the temperature of the dessert, to the nearest hundredth of a degree, after 1 hour and after 2 hours.

Anthropologists and forensic scientists classify skulls using $$ \frac{L+60 W}{L}-\frac{L-40 W}{L} $$ where \(L\) is the skull's length and \(W\) is its width. a. Express the classification as a single rational expression. b. If the value of the rational expression in part (a) is less than \(75,\) a skull is classified as long. A medium skull has a value between 75 and \(80,\) and a round skull has a value over \(80 .\) Use your rational expression from part (a) to classify a skull that is 5 inches wide and 6 inches long.
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