Solve the following equations:
$x^{4} + 2 x^{3} - 7 x^{2} - 8 x + 12 = 0$ .

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Solution

Given equation, $x^{4} + 2 x^{3} - 7 x^{2} - 8 x + 12 = 0$

Consider $f (x) = x^{4} + 2 x^{3} - 7 x^{2} - 8 x + 12$

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

$∴ f (x) = (x - 1) \cdot g (x) ⟹ g (x) = x^{3} + 3 x^{2} - 4 x - 12$

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

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

$∴$ the roots of the given equation are $x = 1, 2, - 2, - 3$

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Similar questions

x^{4} + 2 x^{3} - 7 x^{2} - 8 x + 12

x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 12

Using factor theorem,factorize the polynomial:

1) x⁴-2x³-7x²+8x+12

x^{4} + 2 x^{3} - 7 x^{2} - 8 x + 12; - 3, - 2, 1, 2

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