Problem 153 - College Algebra Essentials

Question

Perform the indicated operation or operations. $$(2 x-1)\left(x^{2}+x-2\right)$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(2x^{3} + x^{2} - 5x + 2\).

Step 1: Distribute the First Term

First distribute the term 2x in the binomial (2x - 1) to each term in the trinomial \(x^{2} + x - 2\). This results in \(2x*x^{2} + 2x*x - 2*2x\), which simplifies to \(2x^{3} + 2x^{2} - 4x\).

Step 2: Distribute the Second Term

Next distribute the term -1 in the binomial (2x - 1) to each term in the trinomial \(x^{2} + x - 2\). This results in \(-1*x^{2} - 1*x + 2*-1\), which simplifies to \(-x^{2} - x + 2\).

Step 3: Combine Like Terms

Now combine the results from step 1 and step 2 to gain the final solution. This results in \(2x^{3} + 2x^{2} - 4x - x^{2} - x + 2\), which can then be simplified by combining like terms to \(2x^{3} + x^{2} - 5x + 2\).