Problem 18 - College Algebra Essentials

Question

Find each product. $$(2 x-1)\left(x^{2}-4 x+3\right)$$

Step-by-Step Solution

Answer

\(2x^3 - 9x^2 + 10x - 3\)

Step 1: Distribute the first term of the binomial

Multiply the first term of the binomial, which is 2x, with each term of the trinomial. This yields \(2x * x^2 = 2x^3\), \(2x * -4x = -8x^2\) and \(2x * 3 = 6x\). Therefore, the first part of the result is \(2x^3 - 8x^2 + 6x\).

Step 2: Distribute the second term of the binomial

Multiply the second term of the binomial, which is -1, with each term of the trinomial. This yields \(-1 * x^2 = -x^2\), \(-1 * -4x = 4x\) and \(-1 * 3 = -3\). Therefore, the second part of the result is \(-x^2 + 4x - 3\).

Step 3: Combine the results

Now simply write the results from the distribution of each term of the binomial next to each other, remembering to add an addition sign between them. This yields the final product as \(2x^3 - 8x^2 + 6x - x^2 + 4x - 3\).

Step 4: Simplify the final result

Finally, the result can be simplified by combining like terms to obtain the final simplified product of \(2x^3 - 9x^2 + 10x - 3\).