Question

Find the product of the rational roots of the equation ${2x}^4+x^3-{11x}^2+x+2=0$.

Answer 1

Since $x=0$ is not a solution of the given equation.
Dividing by $x^2$ in both sides of the equation, we get
$2(x^2+\frac{1}{x^2})+(x+\frac{1}{x})-11=0$ ..(1)
Put $x+\frac{1}{x}=y$ in equation (1), then equation (1) is reduced in the form
$2(y^2-2)+y-11=0$
$\Rightarrow {2y}^2+y-15=0$
$\Rightarrow y_1=-3$ and $y_2=\frac{5}{2}$
Consequently, the original equation is equivalent to the collection of equations$\{\begin{array}{c}x+\frac{1}{x}=-3\\ x+\frac{1}{x}=\frac{5}{2}\end{array}$
$\Rightarrow x_1=\frac{-3-\sqrt{5}}{2},x_2=\frac{-3+\sqrt{5}}{2},x_3=\frac{1}{2},x_4=2$
So, the product of the roots $=1$.