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Chapter 0: Problem 13

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

Short Answer

Expert verified
The factored form of the expression \(x^{3}-x^{2}+2 x-2\) is \((x-1)(x^{2}+2)\)

Step by step solution

01

Group the terms

The expression to factor by grouping is \(x^{3}-x^{2}+2 x-2\). Group the first two terms together and the last two terms together to get \((x^{3}-x^{2})+(2 x-2)\).
02

Factor out a common factor for each group

For the first group \((x^{3}-x^{2})\), the common factor is \(x^{2}\). Factoring this out gives \(x^{2}(x-1)\). For the second group \((2 x-2)\), the common factor is 2. Factoring this out gives \(2(x-1)\). The expression now looks like this: \(x^{2}(x-1)+2(x-1)\).
03

Factor out the common polynomial

Now, notice that \((x-1)\) is a common factor in both terms. Factoring this out gives \((x-1)(x^{2}+2)\). This is the final factored form of the expression.

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