Question

Rational roots of the equation 2x 4 +x 3 −11x 2 + x+2=0 are

• A. 2 1 ​ ,2,3,4
• B. 2 1 ​ ,2, 4 3 ​ ,−2
• C. 2 1 ​ ,2, 4 1 ​ ,−2
• D. 2 1 ​ and 2

Answer 1

Answer: (D)

2 1 ​ and 2

Answer 2

The given equation $2x^4 + x^3 - 11x^2 + x + 2 = 0$ is a reciprocal equation. Dividing by $x^2$ and substituting $y = x + \frac{1}{x}$ transforms the equation into a quadratic in $y$: $2y^2 + y - 15 = 0$. Factoring this quadratic yields $y = \frac{5}{2}$ or $y = -3$. Substituting back $x + \frac{1}{x}$ for $y$, we solve for $x$. The case $y = \frac{5}{2}$ gives $x + \frac{1}{x} = \frac{5}{2}$, which leads to the quadratic equation $2x^2 - 5x + 2 = 0$. Its roots are $x = \frac{1}{2}$ and $x = 2$, which are rational. The case $y = -3$ gives $x + \frac{1}{x} = -3$, leading to $x^2 + 3x + 1 = 0$, whose roots $x = \frac{-3 \pm \sqrt{5}}{2}$ are irrational. Therefore, the rational roots are $\frac{1}{2}$ and $2$.