Determine the square root of the polynomial: $x^{4} + 4 x^{3} + 8 x^{2} - 8 x + 4$ using long division method.

| x + 2 x + 2 |

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∣ ∣ x^{2} + x + 2 ∣ ∣

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x^{2} + 2 x + 2

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∣ ∣ x^{2} + 2 x + 2 ∣ ∣

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Solution

The correct option is C $∣ ∣ x^{2} + 2 x + 2 ∣ ∣$
$\div$ $x^{2} + 2 x + 2$ <= Quotient
$x^{2}$
$x^{4} + 4 x^{3} + 8 x^{2} - 8 x + 4$
$x^{4}$
$2 x^{2} + 2 x$ $4 x^{3} + 8 x^{2}$
$4 x^{3} + 4 x^{2}$
$2 x^{2} + 4 x + 2$ $4 x^{2} + 8 x + 8$
$4 x^{2} + 8 x + 8$
Remainder $\Rightarrow 0$
The square root of $x^{4} + 4 x^{3} + 8 x^{2} - 8 x + 4$ is $∣ ∣ x^{2} + 2 x + 2 ∣ ∣$

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