Answer :

We start with the expression

[tex]$$
1 - 8x^6.
$$[/tex]

Notice that the term [tex]$8x^6$[/tex] can be written as [tex]$(2x^2)^3$[/tex] because

[tex]$$
(2x^2)^3 = 8x^6.
$$[/tex]

Thus, we have

[tex]$$
1 - 8x^6 = 1 - (2x^2)^3.
$$[/tex]

This expression is now a difference of cubes. Recall the formula for the difference of cubes:

[tex]$$
a^3 - b^3 = (a - b)(a^2 + ab + b^2).
$$[/tex]

Here we can set

[tex]$$
a = 1 \quad \text{and} \quad b = 2x^2.
$$[/tex]

Substitute these into the formula:

1. The first factor is

[tex]$$
a - b = 1 - 2x^2.
$$[/tex]

2. The second factor is

[tex]$$
a^2 + ab + b^2 = 1^2 + 1\cdot (2x^2) + (2x^2)^2 = 1 + 2x^2 + 4x^4.
$$[/tex]

Thus, the factorization becomes

[tex]$$
1 - 8x^6 = (1 - 2x^2)(1 + 2x^2 + 4x^4).
$$[/tex]

In many cases, it is customary to express the factor with a positive leading coefficient. Notice that

[tex]$$
1 - 2x^2 = -(2x^2 - 1).
$$[/tex]

Substituting this back into the factorization gives

[tex]$$
1 - 8x^6 = -(2x^2 - 1)(1 + 2x^2 + 4x^4).
$$[/tex]

To write the answer neatly:

[tex]$$
1 - 8x^6 = -(2x^2 - 1)(4x^4 + 2x^2 + 1).
$$[/tex]

This is the fully factorized form of the original expression.

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