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Chapter 0: Problem 12

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

Short Answer

Expert verified
The factored form of \(x^{3} - 3 x^{2} + 4 x - 12\) by grouping is \((x - 3)(x^2 + 4)\).

Step by step solution

01

Separate the polynomial into two groups

The polynomial can be separated into two groups as follows: \(x^{3} - 3 x^{2}\) and \(4 x - 12\). It is important to note that there're minus signs kept in front of \(3x^{2}\) and \(4x\) respectively.
02

Factor each Group

In the first group \(x^{3} - 3 x^{2}\), \(x^2\) is common so it can be factored out, the result is \(x^2 (x - 3)\). Similarly, in the second group \(4 x - 12\), 4 is the common factor so it can be factored out, and the result is \(4 ( x - 3)\). After factoring, the expression becomes \(x^2 (x - 3) + 4 (x - 3)\).
03

Factor out the Common Binomial

Now, notice that the binomial \((x - 3)\) is common to both terms, factor it out. The expression becomes \((x - 3)(x^2 + 4)\).

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