Find roots of polynomial $$ p(x) = x^5+\frac{5}{2}x^4-2x^3-6x^2+x^2+2 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= 0.6316\\[1 em]x_2 &= 1.2993\\[1 em]x_3 &= -2.5662\\[1 em]x_4 &= -0.9323+0.2838i\\[1 em]x_5 &= -0.9323-0.2838i \end{aligned} $$Explanation
Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 2 } $.
$$ \begin{aligned} x^5+\frac{5}{2}x^4-2x^3-6x^2+x^2+2 & = 0 ~~~ / \cdot \color{blue}{ 2 } \\[1 em] 2x^5+5x^4-4x^3-12x^2+2x^2+4 & = 0 \end{aligned} $$Step 2:
Combine like terms:
$$ 2x^5+5x^4-4x^3 \color{blue}{-12x^2} + \color{blue}{2x^2} +4 = 2x^5+5x^4-4x^3 \color{blue}{-10x^2} +4 $$Step 3:
Polynomial $ 2x^5+5x^4-4x^3-10x^2+4 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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