Using long division method, divide the polynomial $x^{4} - 16 x b y x + 2$

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

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

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x^{3} - 2 x^{2} - 4 x - 8

No worries! We‘ve got your back. Try BYJU‘S free classes today!

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

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Solution

The correct option is A $x^{3} - 2 x^{2} + 4 x - 8$
Let us first factorize $x^{4} - 16$ as shown below:

$x^{4} - 16 = {(x^{2})}^{2} - 4^{2} = (x^{2} + 4) (x^{2} - 4) (∵ a^{2} - b^{2} = (a + b) (a - b)) = (x^{2} + 4) (x^{2} - 2^{2}) = (x^{2} + 4) (x + 2) (x - 2) . . . . . (1)$

Now let us divide equation 1 by $(x + 2)$ as follows:

$\frac{(x^{2} + 4) (x + 2) (x - 2)}{(x + 2)} = (x^{2} + 4) (x - 2) = x^{3} - 2 x^{2} + 4 x - 8$

Hence, when the polynomial $x^{4} - 16$ is divided by $(x + 2)$ then the result is $x^{3} - 2 x^{2} + 4 x - 8$ .

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