First slide

Theory of equations

Question

The roots of the equation $x^{4} + 3 x^{3} - 3 x - 1 = 0$ are

Moderate

A

$- 1, 1, \frac{- 3 \pm \sqrt{5}}{2}$
B

$1, 1, \frac{- 3 \pm \sqrt{5}}{2}$
C

$- 1, - 1, \frac{- 2 \pm \sqrt{5}}{2}$
D

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

Solution

The given equation is $x^{4} + 3 x^{3} - 3 x - 1 = 0$
By using synthetic division, we get

$1 \begin{matrix} 1 3 0 - 3 - 1 \\ 0 1 4 4 1 \end{matrix}$

$- 1 \begin{matrix} 1 4 4 1 0 \\ 0 - 1 - 3 - 1 \end{matrix}$

$1 3 1 0$

$∴ (x + 1) (x - 1) (x^{2} + 3 x + 1) = 0$

$\Rightarrow (x^{2} - 1) (x^{2} + 3 x + 1) = 0$

$∴ x = \pm 1$ $; x = \frac{- 3 \pm \sqrt{9 - 4}}{2}$

$x = \pm 1, \frac{- 3 \pm \sqrt{5}}{2}$

Hence (1) is correct

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