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Chapter 7: Problem 77

Simplify each rational expression. $$\frac{x^{2}-9 x+18}{x^{3}-27}$$

Short Answer

Expert verified
\(\frac{x-6}{x^2+3x+9}\)

Step by step solution

01

Factor the Numerator

Rewrite the numerator \(x^{2}-9x+18\) as \((x-3)(x-6)\) using factoring techniques for quadratic expressions.
02

Factor the Denominator

Factoring the denominator involves recognizing the difference of cubes formula, which is \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). The denominator \(x^3-27\) can be written as \((x-3)(x^2+3x+9)\)
03

Simplify the Expression

You should cancel any common factors between the numerator and the denominator. In this case, \((x-3)\) appears in both the numerator and the denominator, so it can be cancelled out. This simplifies the expression to \(\frac{x-6}{x^2+3x+9}\).

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Most popular questions from this chapter

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{2 y}{y-5}-\left(\frac{2}{y-5}+\frac{y-2}{y-5}\right)$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{2 x-y}{x-y}+\frac{x-2 y}{y-x}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x-1}{6}+\frac{x+2}{3}$$

perform the indicated operations. Simplify the result, if possible. $$\left(\frac{3 x^{2}-4 x+4}{3 x^{2}+7 x+2}-\frac{10 x+9}{3 x^{2}+7 x+2}\right) \div \frac{x-5}{x^{2}-4}$$

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.
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