Expert: Scott A Wilson - 10/22/2009

Question
Solve 3x^4=24x by factoring and then using the zero-product principle.

I've tried this, and kept getting {0,2,-2}  the correct answer is {0,2,-1+(square root of 3)i, and -1-(square root of 3)i}.  there is a problem like this in my text book, but no examples that I can find.  The substituting x^4 with u^2 doesn't seem possible with the 24x.  Please help!  Thank you!

Answer
The problem needs to have all taken to the left, so it is 3x^4 - 24x = 0.
Note that a 3x can be factored out, giving 3x(x^3 - 8).

This says the solutions are at x=0 for the 3x out front, and where x^3 = 8.

To calculate cube roots, convert the number to polar coordinates.
The number 8, in polar coordinates, is (8,0°).

The way to calculate a cube root is to add 360° to the number 1 less than the root that is being taken.  Since we are computing the cube root, that corresponds to 3, so add 360° on there twice.
That is, use (8,0°), (8,360°), and (8, 720°).  Once this has been done, compute the cube root of the first part and divide the angle by 3.  The cube root of 8 is 2, so the answers are
(2,0°), (2,120°), and (2,240°).

To convert these back to real numbers, know that x = r•cosΘ and y = r•sinΘ.

For the 1st point, (2,0°), that converts to (2,0).
For the 2nd point, (2,120°), that converts to (2•cos(120°), 2•sin(120°)) = (-1, √3i).
For the 3rd point,  (2,240°), that converts to (2•cos(240°), 2•sin(240°)) = (-1, -√3i).