Find roots of polynomial $$ p(x) = \frac{1}{4}x^5+8x+5x^4-19 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -1.6245\\[1 em]x_2 &= 1.1634\\[1 em]x_3 &= -19.9955\\[1 em]x_4 &= 0.2283+1.3996i\\[1 em]x_5 &= 0.2283-1.3996i \end{aligned} $$Explanation
Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 4 } $.
$$ \begin{aligned} \frac{1}{4}x^5+8x+5x^4-19 & = 0 ~~~ / \cdot \color{blue}{ 4 } \\[1 em] x^5+32x+20x^4-76 & = 0 \end{aligned} $$Step 2:
Write polynomial in descending order
$$ \begin{aligned} x^5+32x+20x^4-76 & = 0\\[1 em] x^5+20x^4+32x-76 & = 0 \end{aligned} $$Step 3:
Polynomial $ x^5+20x^4+32x-76 $ has no rational roots that can be found using Rational Root Test, so the roots were found using Newton method.
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