Find roots of polynomial $$ p(x) = 2n^4+n^3 $$

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

$$ \begin{aligned}n_1 &= 0\\[1 em]n_2 &= -\frac{ 1 }{ 2 } \end{aligned} $$

Step 1:

Factor out $ \color{blue}{ n^3 }$ from $ 2n^4+n^3 $ and solve two separate equations:

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

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

Step 2:

To find the last zero, solve equation $ 2n+1 = 0 $

$$ \begin{aligned} 2n+1 & = 0 \\[1 em] 2 \cdot n & = -1 \\[1 em] n & = - \frac{ 1 }{ 2 } \end{aligned} $$

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