Find roots of polynomial $$ p(x) = x^5+16x^4+26x^3+10x^2+x $$

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

$$ \begin{aligned}x_1 &= 0\\[1 em]x_2 &= -0.1603\\[1 em]x_3 &= -0.3443\\[1 em]x_4 &= -1.2747\\[1 em]x_5 &= -14.2208 \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ x }$ from $ x^5+16x^4+26x^3+10x^2+x $ and solve two separate equations:

$$ \begin{aligned} x^5+16x^4+26x^3+10x^2+x & = 0\\[1 em] \color{blue}{ x }\cdot ( x^4+16x^3+26x^2+10x+1 ) & = 0 \\[1 em] \color{blue}{ x = 0} ~~ \text{or} ~~ x^4+16x^3+26x^2+10x+1 & = 0 \end{aligned} $$

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

Step 2:

Polynomial $ x^4+16x^3+26x^2+10x+1 $ has no rational roots that can be found using Rational Root Test, so the roots were found using quartic formulas.

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