What will be the quotient and reminder, if we divide $x^{4} - {8 x}^{2} + 16$ and $x + 2$ using synthetic division?

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({x^{3} - 4 x + 8)}^{}

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

({x^{3} - 2 x^{2} - x + 8)}^{}

({x^{3} - 2 x^{2} + 8)}^{}

answer is B.

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Detailed Solution

According to the question we have to divide

x^{4} - {8 x}^{2} + 16

and

x + 2

with the help of the synthetic division method. That is

\frac{x^{4} - {8 x}^{2} + 16}{x + 2}

.
And here,

x^{4} - {8 x}^{2} + 16

is a dividend while

x + 2

is a divisor.
And the dividend is a 4-degree polymer.
We know that the synthetic division method is used when a polynomial is to be divided by a linear expression in which the coefficient should be equal to 1.
So, First, we expand the dividend that is

x^{4} - {8 x}^{2} + 16

to find its coefficients.

{\Rightarrow x}^{4} + {0 . x}^{3} - {8 x}^{2} + 0 . x + 16

From here, we can say that ith coefficient of

x^{4}, x^{3}, x^{2}, and x are 1, 0, - 8, 0, and 16

respectively.
Now, we have to find the value of the root of the divisor and for that, we write that the divisor is equal to zero. That is

(x + 2) = 0

, and the constant term in the divisor will be the root of

(x + 2) = 0

.
So, the constant term in the divisor is:

(x + 2) = 0

\Rightarrow x = - 2

Now, Bring down the first coefficient of the dividend So, we need to write down the above terms in the following way:

- 2 ⎜ 1, 0, - 8, 0, 16

Now, we bring down the first coefficient of the dividend and write the first term directly as following

- 2 ⎜ \frac{1}{1}, 0, - 8, 0, 16

Now, cross multiply it by -2 and then add it to the above coefficient and proceed with the same process for next all terms

- 2 ⎜ \frac{1}{1} \frac{0}{- 2} \frac{- 8}{4} \frac{0}{8} \frac{16}{- 16}

Now, write the above terms as the coefficient of terms starts with the one degree less from the greatest degree which is given in the question as 4.
So, the term would become