Problem 12 - Algebra and Trigonometry

Question

Factor by grouping. $$x^{3}-3 x^{2}+4 x-12$$

Step-by-Step Solution

Answer

The factored form of the given polynomial is \((x^{2}+4)(x-3).\)

Step 1: Separate the polynomial into two groups

To factor by grouping, first group the terms two by two, like this: \( (x^{3}-3 x^{2}) + (4 x-12) \). Note that we keep the signs within parentheses.

Step 2: Factor out any common factors in each group

The next step is to factor out the largest common factor from each group separately. In the first group, we can factor out \(x^{2}\) to get: \(x^{2}(x-3) \). In the second group, we can factor out 4, yielding: \(4(x-3)\). This gives us: \(x^{2}(x-3) + 4(x-3)\).

Step 3: Factor out the common binomial

In the previous step, we see that both groups share the common binomial \(x-3\). We can treat this entire binomial as a common factor. We thus factor out \( (x-3)\) to get: \((x^{2}+4)(x-3)\).