First slide

Theory of equations

Question

The roots of the equation $2 x^{4} + x^{3} - 6 x^{2} + x + 2 = 0$ are

Moderate

A

$\frac{1}{2}, 1, 1, 2$
B

$1, 1, - 2 - \frac{1}{2}$
C

$1, 1, 2, - \frac{1}{2}$
D

$1, 1, \frac{3 + \sqrt{5}}{2}$

Solution

the given equation is $2 x^{4} + x^{3} - 6 x^{2} + x + 2 = 0$
By using Synthetic division, we get
$1 \begin{matrix} 2 1 - 6 1 2 \\ 0 2 3 - 3 - 2 \end{matrix}$
   $2 3 - 3 - 2$    $0$

$\begin{matrix} 1 \begin{matrix} 2 3 - 3 - 2 0 \\ 0 2 5 2 \end{matrix} \\ 2   5     2       0 \end{matrix}$

$\Rightarrow {(x - 1)}^{2} (2 x^{2} + 5 x + 2) = 0$
$\Rightarrow {(x - 1)}^{2} (x + 2) (2 x + 1) = 0$
$x = 1, 1, - 2, \frac{- 1}{2}$
Hence (2) is correct choice

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