How do you find all the zeros of x^5+x^3-30x ?

Jul 8, 2018

Answer:

Factor to find zeros:

$0 , \pm \sqrt{5}$ and $\pm \sqrt{6} i$

Explanation:

After separating out the common factor $x$, the remaining quartic can be factored like a quadratic, before reducing to linear factors:

${x}^{5} + {x}^{3} - 30 x = x \left({x}^{4} + {x}^{2} - 30\right)$

$\textcolor{w h i t e}{{x}^{5} + {x}^{3} - 30 x} = x \left({x}^{2} + 6\right) \left({x}^{2} - 5\right)$

$\textcolor{w h i t e}{{x}^{5} + {x}^{3} - 30 x} = x \left(x - \sqrt{6} i\right) \left(x + \sqrt{6} i\right) \left(x - \sqrt{5}\right) \left(x + \sqrt{5}\right)$

Hence zeros:

$0 , \pm \sqrt{5}$ and $\pm \sqrt{6} i$