Question

which shows one way to determine the factors of $x^3 - 12x^2 - 2x + 24$ by grouping?\
\\(\bigcirc\\ x(x^2 - 12) + 2(x^2 - 12)\\)\
\\(\bigcirc\\ x(x^2 - 12) - 2(x^2 - 12)\\)\
\\(\bigcirc\\ x^2(x - 12) + 2(x - 12)\\)\
\\(\bigcirc\\ x^2(x - 12) - 2(x - 12)\\)

Explanation:

Step1: Group the terms

The polynomial is \(x^{3}-12x^{2}-2x + 24\). Group the first two terms and the last two terms: \((x^{3}-12x^{2})+(-2x + 24)\).

Step2: Factor out GCF from each group

For the first group \(x^{3}-12x^{2}\), the greatest common factor (GCF) is \(x^{2}\). Factoring out \(x^{2}\) gives \(x^{2}(x - 12)\).

For the second group \(-2x + 24\), we can factor out \(- 2\) (or factor out \(2\) and notice the sign). Factoring out \(-2\) gives \(-2(x - 12)\) (which is equivalent to \(2(-x + 12)\), but we want a common binomial factor with the first group, so factoring out \(-2\) gives \(-2(x - 12)\)). So the grouped and factored form is \(x^{2}(x - 12)-2(x - 12)\).

Answer:

\(x^{2}(x - 12)-2(x - 12)\) (the fourth option)