Solve the following equations:
$x^{4} - 2 x^{3} - 12 x^{2} + 10 x + 3 = 0$ .

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Solution

Given equation, $x^{4} - 2 x^{3} - 12 x^{2} + 10 x + 3 = 0$

Consider $f (x) = x^{4} - 2 x^{3} - 12 x^{2} + 10 x + 3$

By inspection we can easily see that $f (1) = 0$ . Therefore, $(x - 1)$ is one factor of the given equation

$∴ f (x) = (x - 1) \cdot g (x) ⟹ g (x) = x^{3} - x^{2} - 13 x - 3$

We need to find root of $g (x) = 0$

$⟹ x^{3} - x^{2} - 13 x - 3 = 0 ⟹ x^{3} + 3 x^{2} - 4 x^{2} - 12 x - x - 3 = 0 ⟹ (x + 3) (x^{2} - 4 x - 1) = 0$

Solving the quadratic equations, we have $x = 2 \pm \sqrt{5}$

$∴$ The roots of the given equation are $x = 1, - 3, 2 \pm \sqrt{5}$

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