Question

Factor completely 16x8 − 1.

(4x4 − 1)(4x4 + 1)
(2x2 − 1)(2x2 + 1)(4x4 + 1)
(2x2 − 1)(2x2 + 1)(2x2 + 1)(2x2 + 1)
(2x2 − 1)(2x2 + 1)(4x4 − 1)

Answer 1

Given:

The given expression is

We need to determine the factor of the given expression.

Factor:

Let us rewrite the given expression.

Thus, we have;

Since, the above expression is of the form , let us apply the identity

Thus, we have;

------ (1)

Now, we shall factor the term

Substituting the above expression in equation (1), we have;

Therefore, the factor of the given expression is

Hence, Option B is the correct answer.

Answer 2

Given:

The given expression is
`16 x^(8)-1`

We need to determine the factor of the given expression.

Factor:

Let us rewrite the given expression.

Thus, we have;

`\left(4 x^(4)\right)^(2)-1^(2)`

Since, the above expression is of the form
`a^2-b^2`, let us apply the identity
`a^2-b^2=(a+b)(a-b)`

Thus, we have;

`\left(4 x^(4)+1\right)\left(4 x^(4)-1\right)` ------ (1)

Now, we shall factor the term
`4x^4-1`

`4x^4-1=(2x^2)^2-1^2`

`=(2x^2+1)(2x^2-1)`

Substituting the above expression in equation (1), we have;

`\left(4 x^(4)+1\right)\left(2 x^(2)+1\right)(2x^2-1)`

Therefore, the factor of the given expression is
`\left(4 x^(4)+1\right)\left(2 x^(2)+1\right)(2x^2-1)`

Hence, Option B is the correct answer.