Find roots of polynomial $$ p(x) = 4x^8-1+6x^6-3x^9 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -0.6942\\[1 em]x_2 &= 0.7238\\[1 em]x_3 &= 1.8908\\[1 em]x_4 &= 0.3947+0.6161i\\[1 em]x_5 &= 0.3947-0.6161i\\[1 em]x_6 &= -0.4502+0.6266i\\[1 em]x_7 &= -0.4502-0.6266i\\[1 em]x_8 &= -0.238+1.0219i\\[1 em]x_9 &= -0.238-1.0219i \end{aligned} $$Explanation
Step 1:
Write polynomial in descending order
$$ \begin{aligned} 4x^8-1+6x^6-3x^9 & = 0\\[1 em] -3x^9+4x^8+6x^6-1 & = 0 \end{aligned} $$Step 2:
Polynomial $ -3x^9+4x^8+6x^6-1 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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