Question

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

( x(x^2 - 12) + 2(x^2 - 12) )
( x(x^2 - 12) - 2(x^2 - 12) )
( x^2(x - 12) + 2(x - 12) )
( x^2(x - 12) - 2(x - 12) )

Explanation:

Step1: Analyze the original polynomial

The original polynomial is \(x^3 - 12x^2 - 2x + 24\). We want to factor by grouping. First, group the first two terms and the last two terms: \((x^3 - 12x^2)+(-2x + 24)\).

Step2: Factor out the GCF from each group

For the first group \(x^3 - 12x^2\), the greatest common factor (GCF) is \(x^2\), so factoring out \(x^2\) gives \(x^2(x - 12)\). For the second group \(-2x + 24\), the GCF is \(-2\), so factoring out \(-2\) gives \(-2(x - 12)\).

Step3: Combine the factored groups

Putting it together, we have \(x^2(x - 12)-2(x - 12)\), which matches the fourth option.

Answer:

\(x^2(x - 12)-2(x - 12)\) (the fourth option: \(x^2(x - 12)-2(x - 12)\))