Problem 13 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \((x^{3}-x^{2})+(2 x-2)\).

Step 2: 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: \(x^{2}(x-1)+2(x-1)\).

Step 3: 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.