Find roots of polynomial $$ p(x) = 2x^4+5x^3+3x^2+8x+\frac{12}{2}x+3 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -0.2213\\[1 em]x_2 &= -2.7919\\[1 em]x_3 &= 0.2566+1.5371i\\[1 em]x_4 &= 0.2566-1.5371i \end{aligned} $$Explanation
Step 1:
Get rid of fractions by multipling equation by $ \color{blue}{ 2 } $.
$$ \begin{aligned} 2x^4+5x^3+3x^2+8x+\frac{12}{2}x+3 & = 0 ~~~ / \cdot \color{blue}{ 2 } \\[1 em] 4x^4+10x^3+6x^2+16x+12x+6 & = 0 \end{aligned} $$Step 2:
Combine like terms:
$$ 4x^4+10x^3+6x^2+ \color{blue}{16x} + \color{blue}{12x} +6 = 4x^4+10x^3+6x^2+ \color{blue}{28x} +6 $$Step 3:
Polynomial $ 4x^4+10x^3+6x^2+28x+6 $ has no rational roots that can be found using Rational Root Test, so the roots were found using quartic formulas.
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