Find roots of polynomial $$ p(x) = x^6+x^4-9x^5+2x^3-7x^3-7x+8 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0.71\\[1 em]x_2 &= 8.9516\\[1 em]x_3 &= 0.4085+0.9929i\\[1 em]x_4 &= 0.4085-0.9929i\\[1 em]x_5 &= -0.7393+0.7386i\\[1 em]x_6 &= -0.7393-0.7386i \end{aligned} $$Explanation
Step 1:
Combine like terms:
$$ x^6+x^4-9x^5+ \color{blue}{2x^3} \color{blue}{-7x^3} -7x+8 = x^6-9x^5+x^4 \color{blue}{-5x^3} -7x+8 $$Step 2:
Polynomial $ x^6-9x^5+x^4-5x^3-7x+8 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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