Find roots of polynomial $$ p(x) = 9x^4-3x^5+5x^2+17 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 3.2144\\[1 em]x_2 &= 0.6508+1.0005i\\[1 em]x_3 &= 0.6508-1.0005i\\[1 em]x_4 &= -0.7579+0.8144i\\[1 em]x_5 &= -0.7579-0.8144i \end{aligned} $$Explanation
Step 1:
Write polynomial in descending order
$$ \begin{aligned} 9x^4-3x^5+5x^2+17 & = 0\\[1 em] -3x^5+9x^4+5x^2+17 & = 0 \end{aligned} $$Step 2:
Polynomial $ -3x^5+9x^4+5x^2+17 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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