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What will be the factors of $x 3 + 6 x 2 + 12 x + 16$ ?

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(x + 4) x 2 + 2 x + 4

(x − 4) x 2 + 2 x + 4

(x + 6) x 2 + 2 x + 4

(x + 8) x 2 + 2 x + 4

answer is A.

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Detailed Solution

We have to find the factors of

x 3 + 6 x 2 + 12 x + 16

.
Evaluating one of the three zeros of the expression using the method of trial and error, Replacing x with

± 1, ± 2, ± 3, ± 4

until the value of the expression is 0.
For x = (-4),

x 3 + 6 x 2 + 12 x + 16 = (− 4) 3 + 6 (− 4) 2 + 12 (− 4) + 16 x 3 + 6 x 2 + 12 x + 16 = − 64 + 96 − 48 + 16 x 3 + 6 x 2 + 12 x + 16 = 0

So, one of the zeros is (-4), and subsequently, (x + 4) is a factor.
On dividing the given expression with (x+4), we get the factorization of

x 3 + 6 x 2 + 12 x + 16

(x + 4) x 2 + 2 x + 4

.
The discriminant value of

x 2 + 2 x + 4

22 − 4.4 = − 12

which is negative.
So,

x 2 + 2 x + 4

does not have real zeros.
So, the factors of

x 3 + 6 x 2 + 12 x + 16

are

(x + 4) x 2 + 2 x + 4

.
Therefore, option 1 is correct.

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