Problem 17 - College Algebra

Question

Find each product. $$(2 x-3)\left(x^{2}-3 x+5\right)$$

Step-by-Step Solution

Answer

Thus, the product of the given polynomials is \(2x^{3} - 9x^{2} + 19x - 15\).

Step 1: Distribute the First Term of the First Polynomial

First, distribute the first term of the first polynomial, \(2x\), to each term in the second polynomial. This results in \(2x * x^{2} = 2x^{3}\), \(2x * -3x = -6x^{2}\), and \(2x * 5 = 10x\).

Step 2: Distribute the Second Term of the First Polynomial

Next, distribute the second term of the first polynomial, \(-3\), to each term in the second polynomial. This results in \(-3 * x^{2} = -3x^{2}\), \(-3 * -3x = 9x\) and \(-3 * 5 = -15\).

Step 3: Combine Like Terms

Combine the terms from steps 1 and 2 that are similar (have the same variables and exponents). Thus, \(2x^{3}\) has no similar term, \(-6x^{2}\) and \(-3x^{2}\) add to \(-9x^{2}\), and \(10x\) and \(9x\) add to \(19x\). The constant \(-15\) remains as it is.