Factorise \( \left(x^{3}-1\right) \)

To factor \( x^3 - 1 \), we can use the difference of cubes formula, which states that \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). Here, we can set \( a = x \) and \( b = 1 \). Applying the formula gives: \[ x^3 - 1 = (x - 1)(x^2 + x \cdot 1 + 1^2) = (x - 1)(x^2 + x + 1). \] So, the complete factorization of \( x^3 - 1 \) is \( (x - 1)(x^2 + x + 1) \). To check your work, you can expand the factors back together. Multiply \( (x - 1) \) by \( (x^2 + x + 1) \), and you should get back your original polynomial \( x^3 - 1 \). Just remember, it's easy to slip up with signs or terms, so take your time!