Find roots of polynomial $$ p(x) = 40+24x-6x^2-14x^3-x^4+2x^5-43x^6+x^7 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -0.8913\\[1 em]x_2 &= 1.0106\\[1 em]x_3 &= 42.9542\\[1 em]x_4 &= 0.5142+0.9873i\\[1 em]x_5 &= 0.5142-0.9873i\\[1 em]x_6 &= -0.5509+0.7286i\\[1 em]x_7 &= -0.5509-0.7286i \end{aligned} $$Explanation
Step 1:
Write polynomial in descending order
$$ \begin{aligned} 40+24x-6x^2-14x^3-x^4+2x^5-43x^6+x^7 & = 0\\[1 em] x^7-43x^6+2x^5-x^4-14x^3-6x^2+24x+40 & = 0 \end{aligned} $$Step 2:
Polynomial $ x^7-43x^6+2x^5-x^4-14x^3-6x^2+24x+40 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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