Find roots of polynomial $$ p(x) = x^5+2x^4+3x^4+6x^2+5x+3 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -5.1911\\[1 em]x_2 &= -0.4239+0.4584i\\[1 em]x_3 &= -0.4239-0.4584i\\[1 em]x_4 &= 0.5194+1.1013i\\[1 em]x_5 &= 0.5194-1.1013i \end{aligned} $$Explanation
Step 1:
Combine like terms:
$$ x^5+ \color{blue}{2x^4} + \color{blue}{3x^4} +6x^2+5x+3 = x^5+ \color{blue}{5x^4} +6x^2+5x+3 $$Step 2:
Polynomial $ x^5+5x^4+6x^2+5x+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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