Solve : $x^{3} + 3 x^{2} + 3 x + 7 = 0$

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Solution

$x^{3} - 3 x^{2} + 3 x + 7 = 0$

if we put $x = - 1$ in above equation, we get

$- 1 - 3 - 3 + 7 = - 7 + 7 = 0 \Rightarrow (x + 1)$ is a factor of $x^{3} - 3 x^{2} + 3 x + 7 = 0$

If we divide $x^{3} - 3 x^{2} + 3 x + 7 = 0$ by $(x + 1)$ we will get remaining factors.

$\frac{x^{3} - 3 x^{2} + 3 x + 7}{x + 1} = x^{2} - 4 x + 7$

Solving the quadratic equation $x^{2} - 4 x + 7$ we get,

$= \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

$= \frac{4 \pm \sqrt{16 - 4 \times 1 \times 7}}{2 \times 1} = 2 \pm i \sqrt{3}$

$∴ x = - 1, 2 \pm i \sqrt{3}$ are the factors of $x^{3} - 3 x^{2} + 3 x + 7 = 0$

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