Problem 15 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The expression \(3x^3 - 2x^2 - 6x + 4\) factored by grouping is \((3x - 2)(x^2 - 2)\).

Step 1: Grouping Terms

Separate the given polynomial \(3x^3 - 2x^2 - 6x + 4\) into two groups: \((3x^3 - 2x^2)\) and \((-6x + 4)\).

Step 2: Factoring out the GCF

Factor out the GCF from each group. For the first group \((3x^3 - 2x^2)\), factor out \(x^2\) to get \(x^2(3x - 2)\). For the second group, factor out \(-2\) to get \(-2(3x - 2)\). Now your expression should look like this: \(x^2(3x - 2) - 2(3x - 2)\).

Step 3: Factoring by Grouping

As we see, both groups have a common binomial factor \((3x - 2)\). Factor that out to obtain the factored form of the original expression: \((3x - 2)(x^2 - 2)\).