Problem 23 - College Algebra

Question

Find each product. $$(3 x+5)(2 x+1)$$

Step-by-Step Solution

Answer

The product of the two binomials \(3x+5\) and \(2x+1\) is \(6x^2 + 13x + 5\).

Step 1: Distribute the first term of the first binomial

Distribute the first term (3x) of the first binomial (3x + 5) over the second binomial (2x + 1). Apply the distributive rule: a*(b+c) = a*b + a*c. Here, a is 3x, b is 2x and c is 1. This results in (3x * 2x) + (3x * 1). Calculate the value of these operations to get \(6x^2\) and \(3x\).

Step 2: Distribute the second term of the first binomial

Next, distribute the second term (5) of the first binomial (3x + 5) over the second binomial (2x + 1). Using the distributive rule again, this results in (5 * 2x) + (5 * 1). Calculate these operations to get \(10x\) and \(5\).

Step 3: Combine like terms

After performing steps 1 and 2, four terms are obtained: \(6x^2, 3x, 10x\) and \(5\). Combine the terms that have the same variable, i.e. \(3x\) and \(10x\). This results in \(6x^2 + 13x + 5\).