To factor the polynomial [tex]\( x^3 - 1 \)[/tex], we can use a standard factoring technique. Here's the step-by-step solution:
1. Recognize that [tex]\( x^3 - 1 \)[/tex] can be written as a difference of cubes, since it is in the form [tex]\( a^3 - b^3 \)[/tex], where [tex]\( a = x \)[/tex] and [tex]\( b = 1 \)[/tex].
2. Recall the factoring formula for a difference of cubes:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
3. Apply the formula to [tex]\( x^3 - 1^3 \)[/tex]:
[tex]\[
x^3 - 1 = (x - 1)((x)^2 + (x)(1) + (1)^2)
\][/tex]
4. Simplify the expression within the parentheses:
[tex]\[
x^3 - 1 = (x - 1)(x^2 + x + 1)
\][/tex]
So, the factored form of the polynomial [tex]\( x^3 - 1 \)[/tex] is [tex]\((x - 1)(x^2 + x + 1)\)[/tex].
Therefore, the correct answer is:
[tex]\[
(x-1)\left(x^2+x+1\right)
\][/tex]
