# Find all 4 roots of the polynomial: f ( x ) = x 4 </msup> + 2 x

Find all 4 roots of the polynomial:
$f\left(x\right)={x}^{4}+2{x}^{3}-x-1.$
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With the substitution $x=t-b/na$, you eliminate the second highest order term of a polynomial $a{x}^{n}+b{x}^{n-1}+\cdots .$
If you substitute $x=t-2/4=t-1/2$ your polynomial becomes
${t}^{4}-\frac{3}{2}{t}^{2}-\frac{11}{16}.$
This is quadratic in ${t}^{2}$ so you can solve to get
${t}^{2}=\frac{3±2\sqrt{5}}{4}.$
So the four roots of your polynomial are plus and minus the square roots of those two solutions (plus 1/2 because of the original substitution.) We were lucky that the first order term also disappeared.