Problem 83 - Introductory Algebra for College Students

Question

Factor by grouping. $$3 x^{3}-2 x^{2}-6 x+4$$

Step-by-Step Solution

Answer

The factored form of the given polynomial by grouping method is \((3x - 2)(x^2 - 2)\)

Step 1: Group the Polynomial

Firstly, the given polynomial \(3x^3 - 2x^2 - 6x + 4\) can be divided into two groups: \(3x^3 - 2x^2\) and \(-6x + 4\).

Step 2: Factor out the greatest common factor from each group

We factor out the GCF of each group separately. The GCF of \(3x^3 - 2x^2\) is \(x^2\), so we factor it out from the first group, and the GCF of \(-6x + 4\) is \(-2\), so we factor this one out from the second group. This gives us \(x^2(3x - 2) - 2(3x - 2)\).

Step 3: Factor by Grouping

Now we can see that \(3x - 2\) is a common factor in both terms, so we group them together as: \((3x - 2)(x^2 - 2)\).