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Chapter 3: Problem 46

Solve the equation \(3 x^{3}+7 x^{2}-22 x-8=0\) given that \(-\frac{1}{3}\) is a root.

Short Answer

Expert verified
By following these steps for the given cubic equation, it can be simplified to a quadratic equation using synthetic division. The solutions can then be found by solving the quadratic equation.

Step by step solution

01

Verify the given root

Firstly, it would be necessary to substitute the given root \(-\frac{1}{3}\) into the equation to verify whether it鈥檚 indeed a root or not. If the resulting value is zero, it means it鈥檚 a root.
02

Synthetic division

If the provided root was correct, the next step would be to perform synthetic division by dividing the cubic equation by \(x - (-1/3)\), i.e., \(x + \frac{1}{3}\). This will reduce the cubic equation to a quadratic equation.
03

Solve the quadratic equation

After step 2, we should be left with a quadratic equation of form \(ax^{2}+bx+c=0\). We solve this quadratic equation to find the other two remaining roots. If it can鈥檛 be factored easily, use the Quadratic formula to solve it: \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\).
04

Verify each root

The last step is to confirm each root by substituting them back on the original cubic equation to make sure they satisfy it.

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Most popular questions from this chapter

Exercises will help you prepare for the material covered in the next section. If \(f(x)=x^{3}-2 x-5,\) find \(f(2)\) and \(f(3) .\) Then explain why the continuous graph of \(f\) must cross the \(x\) -axis between 2 and 3.

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{x}{2 x+6}-\frac{9}{x^{2}-9}$$

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. $$y=-4 x^{2}+20 x+160$$

Solve each inequality in Exercises \(65-70\) and graph the solution set on a real number line. $$ \left|x^{2}+6 x+1\right|>8 $$

A company is planning to manufacture mountain bikes. The fixed monthly cost will be \(\$ 100,000\) and it will cost \(\$ 100\) to produce each bicycle. a. Write the cost function, \(C\), of producing \(x\) mountain bikes. b. Write the average cost function, \(C,\) of producing x mountain bikes c. Find and interpret \(C(500), C(1000), C(2000),\) and \(C(4000)\) d. What is the horizontal asymptote for the graph of the average cost function, \(C\) ? Describe what this means in practical terms.
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