To rewrite the expression [tex]\( x^3 - 8 \)[/tex] using the difference of cubes formula, we need to apply the appropriate algebraic formula. The difference of cubes formula is:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
In this problem, we identify [tex]\( a \)[/tex] and [tex]\( b \)[/tex] as follows:
1. Recognize that [tex]\( x^3 \)[/tex] is [tex]\( (x)^3 \)[/tex], so [tex]\( a = x \)[/tex].
2. Recognize that 8 can be written as [tex]\( 2^3 \)[/tex], so [tex]\( b = 2 \)[/tex].
Thus, we apply the difference of cubes formula:
[tex]\[ x^3 - 2^3 = (x - 2)\big(x^2 + x \cdot 2 + 2^2\big) \][/tex]
This simplifies to:
[tex]\[ x^3 - 8 = (x - 2)\big(x^2 + 2x + 4\big) \][/tex]
So, the rewritten expression is:
[tex]\[ (x - 2)\big(x^2 + 2x + 4\big) \][/tex]
Comparing this result with the given options:
A. [tex]\((x+2)\left(x^2-2x+4\right)\)[/tex]
B. [tex]\((x-2)\left(x^2+2x+4\right)\)[/tex]
C. [tex]\((x+2)\left(x^2+2x-4\right)\)[/tex]
D. [tex]\((x-2)\left(x^2-2x+4\right)\)[/tex]
The correct answer is:
[tex]\[ \boxed{B} \][/tex]
This corresponds to [tex]\((x - 2)\left(x^2 + 2x + 4\right)\)[/tex].
