Find roots of polynomial $$ p(x) = x^4-7x^3+18x^2-20x+x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= 3.2757\\[1 em]x_3 &= 1.8622+1.5273i\\[1 em]x_4 &= 1.8622-1.5273i \end{aligned} $$

Step 1:

Combine like terms:

$$ x^4-7x^3+18x^2 \color{blue}{-20x} + \color{blue}{x} = x^4-7x^3+18x^2 \color{blue}{-19x} $$

Step 2:

Factor out $ \color{blue}{ x }$ from $ x^4-7x^3+18x^2-19x $ and solve two separate equations:

$$ \begin{aligned} x^4-7x^3+18x^2-19x & = 0\\[1 em] \color{blue}{ x }\cdot ( x^3-7x^2+18x-19 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ x^3-7x^2+18x-19 & = 0 \end{aligned} $$

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

Step 3:

Polynomial $ x^3-7x^2+18x-19 $ has no rational roots that can be found using Rational Root Test, so the roots were found using qubic formulas.

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