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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
The solutions to the equation \(3 x^{3}+7 x^{2}-22 x-8=0\) are \(-\frac{1}{3}\), \(-\frac{4}{3}\) and -2.

Step by step solution

01

Confirm the root

By substitution of \(-\frac{1}{3}\) into the equation, you can verify that indeed, it is a root of the cubic equation. This means that when \(-\frac{1}{3}\) is substituted for \(x\), the equation becomes 0.
02

Synthetic Division

Now apply synthetic division to factor the cubic equation. Draw a box and write the coefficients of the original polynomial (3, 7, -22, -8) in the top row. Write the given root on the upper left corner outside the box. Multiply this root by each value gotten in the bottom row, writing the result in the next column of the second row, then add this to the number already in the second row. Repeat this until you get to the end of the line. The final result should be 0 confirming that \(-\frac{1}{3}\) is indeed a root. The numbers in the bottom row are the coefficients of the quotient polynomial, which is a quadratic.
03

Solve Quadratic Equation

The division process will yield a quadratic equation \(3x^2 + 8x + 8 = 0\). This can be solved using the quadratic formula \(-\frac{b \pm sqrt{b^2 - 4ac}}{2a}\). This gives the roots which are the solutions to the original cubic equation.

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Most popular questions from this chapter

The rational than \(f(x)=\frac{27,725(x-14)}{x^{2}+9}-5 x\) models the number of arrests, \(f(x)\), per \(100,000\) drivers, for driving under the influence of alcohol, as a function of a driver's age, \(x\). a. Graph the function in a \([0,70,5]\) by \([0,400,20]\) viewing rectangle. b. Describe the trend shown by the graph. c. Use the \(200 \mathrm{M}\) and \(\overline{\mathrm{TRACE}}\), features or the maximum function feature of your graphing utility to find the age that corresponds to the greatest number of arrests. How many arrests, per \(100,000\) drivers, are there for this age group?

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