Problem 76 - Introductory Algebra for College Students

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 \((x^{2}-2)(x+6)\)

Step 1: Group the terms

Group the first two terms together and the last two terms together in the given polynomial: \\((x^{3}+6 x^{2})+(-2 x-12)\)

Step 2: Factor out the GCF from each group

Factor out the GCF from each group. In the first group, \(x^{2}\) can be factored out, and in the second group, \(2\) can be factored out: \\(x^{2}(x+6) - 2(x+6)\)

Step 3: Check if the results from step 2 are the same

The expressions in the parentheses are both \(x+6\). So, \(x+6\) is the common binomial and it can be factored out.

Step 4: Factor out the binomial

Factor out the common binomial \(x+6\) to write the original expression in its factored form: \ ((x^{2}-2)(x+6))