Problem 73 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: Group Terms

First, let's rewrite the polynomial \(x^{3} - 2x^{2} +5x - 10\) and isolate each pair of terms: \((x^{3} - 2x^{2}) + (5x - 10)\).

Step 2: Factor Out GCF from Each Group

Now, let's factor out the GCF from each pair. From \(x^{3} - 2x^{2}\), we can factor out \(x^{2}\), resulting in \(x^{2}(x - 2)\). From \(5x - 10\), we can factor out 5, resulting in \(5(x - 2)\). After doing so, the equation becomes \(x^{2}(x - 2) + 5(x - 2)\).

Step 3: Factor the Resulting Expression

The resulting expression, \(x^{2}(x - 2) + 5(x - 2)\), has a common factor of \(x - 2\). We factor this out to obtain \((x - 2)(x^{2} + 5)\), which is the factored form of the original polynomial.