Find roots of polynomial $$ p(x) = x^9+x^5 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 0.7071+0.7071i\\[1 em]x_3 &= 0.7071-0.7071i\\[1 em]x_4 &= -0.7071+0.7071i\\[1 em]x_5 &= -0.7071-0.7071i \end{aligned} $$Explanation
Step 1:
Factor out $ \color{blue}{ x^5 }$ from $ x^9+x^5 $ and solve two separate equations:
$$ \begin{aligned} x^9+x^5 & = 0\\[1 em] \color{blue}{ x^5 }\cdot ( x^4+1 ) & = 0 \\[1 em] \color{blue}{ x^5 = 0} ~~ \text{or} ~~ x^4+1 & = 0 \end{aligned} $$One solution is $ \color{blue}{ x = 0 } $. Use second equation to find the remaining roots.
Step 2:
Polynomial $ x^4+1 $ has no rational roots that can be found using Rational Root Test, so the roots were found using quartic formulas.
This page was created using
Polynomial Roots Calculator