Expanding the Product of Three Binomials: (2x + 1)(2x - 3)(2x + 5)

In high school mathematics, one of the fundamental operations we learn is the expansion of polynomial expressions. This skill is essential for simplifying complex algebraic expressions and preparing for more advanced mathematical concepts. In this article, we will walk through the process of finding the expansion of the product of three binomials: (2x + 1)(2x - 3)(2x + 5).

Step 1: Understand the Distributive Property

The distributive property states that when we multiply a product by a sum, we can distribute the product to each term in the sum. This means that:

(a + b)(c + d) = ac + ad + bc + bd

In our case, we have three binomials, so we will need to apply the distributive property multiple times.

Step 2: Expand the First Two Binomials

Let's start by expanding the first two binomials:

(2x + 1)(2x - 3) = 2x(2x - 3) + 1(2x - 3) = 4x^2 - 6x + 2x - 3 = 4x^2 - 4x - 3

Step 3: Expand the Product of the First Two Binomials with the Third Binomial

Now, we need to expand the product of the first two binomials with the third binomial:

(4x^2 - 4x - 3)(2x + 5) = 4x^2(2x + 5) - 4x(2x + 5) - 3(2x + 5) = 8x^3 + 20x^2 - 8x^2 - 20x - 6x - 15 = 8x^3 + 12x^2 - 26x - 15

Therefore, the expansion of the product (2x + 1)(2x - 3)(2x + 5) is:

8x^3 + 12x^2 - 26x - 15

This result can be used for further algebraic manipulations, factoring, or simplifying more complex polynomial expressions.