Find roots of polynomial $$ p(x) = 3x^4-16x^3-7x^2+64x-20x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 1.7169\\[1 em]x_3 &= -1.6287\\[1 em]x_4 &= 5.2451 \end{aligned} $$

Step 1:

Combine like terms:

$$ 3x^4-16x^3-7x^2+ \color{blue}{64x} \color{blue}{-20x} = 3x^4-16x^3-7x^2+ \color{blue}{44x} $$

Step 2:

Factor out $ \color{blue}{ x }$ from $ 3x^4-16x^3-7x^2+44x $ and solve two separate equations:

$$ \begin{aligned} 3x^4-16x^3-7x^2+44x & = 0\\[1 em] \color{blue}{ x }\cdot ( 3x^3-16x^2-7x+44 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ 3x^3-16x^2-7x+44 & = 0 \end{aligned} $$

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

Step 3:

Polynomial $ 3x^3-16x^2-7x+44 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

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