Problem 11 - Algebra and Trigonometry

Question

Factor by grouping. $$x^{3}-2 x^{2}+5 x-10$$

Step-by-Step Solution

Answer

The factored form of the given polynomial \(x^{3}-2 x^{2}+5 x-10\) is \((x-2)(x^{2} + 5)\).

Step 1: Arrange the polynomial

The given polynomial is \(x^{3}-2 x^{2}+5 x-10\). Here, the terms do not need to be rearranged since they are in standard form (highest degree to lowest degree).

Step 2: Group the terms

Next, group the terms into two pairs. Here, the grouping will look like this: \((x^{3}-2 x^{2}) + (5 x-10)\).

Step 3: Factor out the common factors

In the first group, \(x^{2}\) is a common factor and in the second group, \(5\) is a common factor. Now, factoring out these common factors, we get \(x^{2}(x-2) + 5(x-2)\).

Step 4: Factor the grouped polynomial

Now, notice that \((x-2)\) is a common factor in both terms. So, we can factor out \((x-2)\), resulting in \((x-2)(x^{2} + 5)\). This is the factored form of the given polynomial.