Find roots of polynomial $$ p(x) = 6x^7-2x^6+4x^5+2x^4+7x^3+6x^3+3 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -0.5931\\[1 em]x_2 &= 0.329+0.5169i\\[1 em]x_3 &= 0.329-0.5169i\\[1 em]x_4 &= -0.7031+0.9138i\\[1 em]x_5 &= -0.7031-0.9138i\\[1 em]x_6 &= 0.8373+0.9942i\\[1 em]x_7 &= 0.8373-0.9942i \end{aligned} $$Explanation
Step 1:
Combine like terms:
$$ 6x^7-2x^6+4x^5+2x^4+ \color{blue}{7x^3} + \color{blue}{6x^3} +3 = 6x^7-2x^6+4x^5+2x^4+ \color{blue}{13x^3} +3 $$Step 2:
Polynomial $ 6x^7-2x^6+4x^5+2x^4+13x^3+3 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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