Problem 46 - 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

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 1: Verify the given root

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

Step 2: 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.

Step 3: 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’t be factored easily, use the Quadratic formula to solve it: \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\).

Step 4: 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.