Problem 14 - Algebra and Trigonometry

Question

Factor by grouping. $$x^{3}+6 x^{2}-2 x-12$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(x^{3}+6 x^{2}-2 x-12\) by grouping is \((x+6)(x^{2}-2)\).

Step 1: Group the Terms

Firstly, divide the polynomial into two groups.\[x^{3}+6 x^{2}\] and \[-2 x-12\].

Step 2: Factor out the Greatest Common Factor

Factor out the greatest common factor (GCF) from each group. For the first group, the GCF is \(x^{2}\), so taking it out we get \(x^{2}(x+6)\). For the second group, the GCF is -2, so taking it out we get \(-2(x+6)\). Now, both groups are expressed as \(x^{2}(x+6) -2(x+6)\).

Step 3: Factor out the Common Binomial

Now, factor the common binomial \(x+6\) from both terms. The factored form will be \((x+6)(x^{2}-2)\).