Find roots of polynomial $$ p(x) = 12x^4-x+3x^7-1 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -0.4629\\[1 em]x_2 &= 0.5962\\[1 em]x_3 &= -1.5915\\[1 em]x_4 &= -0.0765+0.5386i\\[1 em]x_5 &= -0.0765-0.5386i\\[1 em]x_6 &= 0.8055+1.3841i\\[1 em]x_7 &= 0.8055-1.3841i \end{aligned} $$Explanation
Step 1:
Write polynomial in descending order
$$ \begin{aligned} 12x^4-x+3x^7-1 & = 0\\[1 em] 3x^7+12x^4-x-1 & = 0 \end{aligned} $$Step 2:
Polynomial $ 3x^7+12x^4-x-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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