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Chapter 7: Problem 84

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

Short Answer

Expert verified
The simplified rational expression is \(\frac{x}{x+3}\)

Step by step solution

01

Factorize the numerator

To factorize the numerator, the common term \(x\) is taken out: \(x^{3}-3 x^{2}+9 x= x(x^{2}-3 x+9)\)
02

Factorize the denominator

For the denominator, recognise that \(x^{3}+27\) can be written as a sum of cubes: \(x^{3}+27 = x^{3} + 3^3\). Hence, using the formula \(a^{3}+b^{3}= (a+b)(a^{2}-ab+b^{2})\), we get \(x^{3} + 27 = (x+3)(x^{2}-3x+9)\).
03

Cancel out common factors

Now the expression becomes \(\frac{x(x^{2}-3 x+9)}{(x+3)(x^{2}-3x+9)}\). The term \(x^{2} - 3x + 9\) in the numerator and the denominator cancels out, therefore the simplified version would be \(\frac{x}{x + 3}\)

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Most popular questions from this chapter

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.

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{6 x+7}{x-6}+\frac{3 x}{6-x}$$

Will help you prepare for the material covered in the next section. a. If \(y=\frac{k}{x},\) find the value of \(k\) using \(x=8\) and \(y=12\) b. Substitute the value for \(k\) into \(y=\frac{k}{x}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=3\)

When two people work together to complete a job, describe one factor that can result in more or less time than the time given by the rational equations we have been using.

Use similar triangles to solve. A person who is 5 feet tall is standing 80 feet from the base of a tree. The tree casts an 86 -foot shadow. The person's shadow is 6 feet in length. What is the tree's height? (IMAGE CANNOT COPY)
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