To factor the expression [tex]\( x^3 - 1 \)[/tex], we use the difference of cubes formula, which states:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
For [tex]\( x^3 - 1 \)[/tex], we identify [tex]\( a = x \)[/tex] and [tex]\( b = 1 \)[/tex]. Plugging these into the formula gives:
[tex]\[ x^3 - 1^3 = (x - 1)(x^2 + x \cdot 1 + 1^2) \][/tex]
Simplifying the expression inside the parentheses:
[tex]\[ (x - 1)(x^2 + x + 1) \][/tex]
Therefore, the factored form of [tex]\( x^3 - 1 \)[/tex] is:
[tex]\[ (x-1)(x^2 + x + 1) \][/tex]
Reviewing the given options:
1. [tex]\(\left(x^3-1\right)\left(x^2+x+1\right)\)[/tex]
2. [tex]\((x-1)\left(x^2-x+1\right)\)[/tex]
3. [tex]\((x-1)\left(x^2+x+1\right)\)[/tex]
4. [tex]\(\left(x^3-1\right)\left(x^2+2x+1\right)\)[/tex]
We see that the correct factored form matches option 3.
Thus, the correct answer is:
[tex]\[ \boxed{(x-1)\left(x^2+x+1\right)} \][/tex]
