solving polynomial equations

[dmouthfan2028]

How do you solve \(\displaystyle 4x^3 - 2x = 2\) ?
I've gotten to \(\displaystyle 2x^3 - x - 1 = 0\), but I don't know where to go from there.
I know \(\displaystyle x = 1\) is a solution, but I'd like to know how to solve this without guessing.

EDIT: Using synthetic division, this factors to \(\displaystyle (2x^2 + 2x + 1)(x-1) = 0\).

 

[JuicyBurger]

EDIT: Using synthetic division, this factors to \(\displaystyle (2x^2 + 2x + 1)(x-1) = 0\).

\(\displaystyle x = 1\) is the only REAL solution.

Use the quadratic formula for \(\displaystyle (2x^2 + 2x + 1)\).

\(\displaystyle \frac{-b \pm \sqrt{b^2 -4ac}}{2a}\)

You will get your other two solutions, both imaginary, from doing this.

Hope this helps!

 

[LivingMath]

The best way to solve a cubic equation depends on your level of math knowledge.

In general, your approach is the correct one unless you are ready to get into some pretty serious math. Typically:

1. Set the equation equal to zero
2. Find one root by trial and error
3. Divide out the first root you found
4. Evaluate the other roots mentally or by using the quadratic formula

If you're ready for some more advanced math, you could look at the general solution to the cubic equation http://en.wikipedia.org/wiki/Cubic_function