Question

Solve $2x^4+x^3-11x^2+x+2=0$

Answer 1

Solution:

$2x^4+x^3-11x+x+2=0$

or, $2x^4-4x^3+5x^3-10x^2-x^2+2x-x+2=0$

or, $2x^3(x-2)+5x^2(x-2)-x(x-2)-1(x-2)=0$

or, $(x-2)(2x^3+5x^2-x-1)=0$

or, $(x-2)(2x^3-x^2+6x^2-3x+2x-1)=0$

or, $(x-2)\left\{x^2\right(2x-1)+3x(2x-1+1(2x-1)\left)\right\}=0$

or, $(x-2)(2x-1)(x^2+3x+1)=0$

Now, $x^2+3x+1=0$

or, $x=\frac{-3\pm \sqrt{9-4}}{2}=\frac{-3\pm \sqrt{5}}{2}$

or, $(x-2)(2x-1)(x-\frac{-3+\sqrt{5}}{2})(x-\frac{-3-\sqrt{5}}{2})=0$

Roots of given equation are $2,\frac{1}{2},\frac{-3+\sqrt{5}}{2}$ and $\frac{-3-\sqrt{5}}{2}.$