Problem 42 - College Algebra

Question

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

Step-by-Step Solution

Answer

The roots of the equation are \(-\frac{1}{3}\), \(3\), and \(-8\).

Step 1: Verify the Given Root

The given root \(-\frac{1}{3}\) can be checked by substituting it into the equation and confirming that the result is 0. Plugging \(-\frac{1}{3}\) into the equation \(3 x^{3}+7 x^{2}-22 x-8=0\) gives \(3(-\frac{1}{3})^{3}+7(-\frac{1}{3})^{2}-22(-\frac{1}{3})-8=0\), which simplifies to 0. This verifies that \(-\frac{1}{3}\) is indeed a root.

Step 2: Use Synthetic Division

The given root can be used to conduct synthetic division on the cubic function to reduce it to a quadratic one. From the synthetic division \(3 x^{3}+7 x^{2}-22 x-8\) divided by \(x + \frac{1}{3}\), we obtain \(3x^{2} + 6x - 24\).

Step 3: Solve Quadratic Equation

The reduced equation, a quadratic one, can be solved for its roots using the quadratic formula. Apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) on equation \(3x^{2} + 6x - 24 = 0\) where \(a = 3\), \(b = 6\), and \(c = -24\), to find the remaining roots.