Factor x^3 - 3x^2 - x + 3.

• A. (x - 1)(x + 1)(x - 3)
• B. (x - 1)(x + 1)(x + 3)
• C. (x - 1)(x - 3)(x + 3)
• D. (x + 1)(x - 3)(x + 3)

Answer: (A)

Factoring by grouping and using a difference of squares is the idea here. Group the terms as x^3 - 3x^2 and -x + 3, pulling out a common factor in each group: x^3 - 3x^2 = x^2(x-3) and -x + 3 = -(x-3). This gives (x-3)(x^2 - 1).

Recognize x^2 - 1 as a difference of squares: (x^2 - 1) = (x-1)(x+1). So the full factorization is (x-3)(x-1)(x+1). Since multiplication is commutative, any order of these three factors works, for example (x-1)(x+1)(x-3).

If you expand (x-3)(x-1)(x+1), you get x^3 - 3x^2 - x + 3, which matches the original expression. The other groupings that would place a plus 3 instead of minus 3 change the sign pattern and don’t reconstruct the original polynomial.