Find roots of polynomial $$ p(x) = 3x^4-33x-6x^2+8x $$

The roots of polynomial $ p(x) $ are:

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 2.3538\\[1 em]x_3 &= -1.1769+1.4681i\\[1 em]x_4 &= -1.1769-1.4681i \end{aligned} $$

Step 1:

Combine like terms:

$$ 3x^4 \color{blue}{-33x} -6x^2+ \color{blue}{8x} = 3x^4-6x^2 \color{blue}{-25x} $$

Step 2:

Factor out $ \color{blue}{ x }$ from $ 3x^4-6x^2-25x $ and solve two separate equations:

$$ \begin{aligned} 3x^4-6x^2-25x & = 0\\[1 em] \color{blue}{ x }\cdot ( 3x^3-6x-25 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ 3x^3-6x-25 & = 0 \end{aligned} $$

One solution is $ \color{blue}{ x = 0 } $. Use second equation to find the remaining roots.

Step 3:

Polynomial $ 3x^3-6x-25 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

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