Problem 53 - College Algebra

Question

Find each product. $$(2 x+3)^{3}$$

Step-by-Step Solution

Answer

The product of `(2x + 3)^3` is `8x^3 + 36x^2 + 54x + 27`.

Step 1: Identify the Components

In our case, we can identify `a` as `2x` and `b` as `3`. We are given that `n` is `3` as the power of the binomial. Now we substitute our identified values into the binomial theorem.

Step 2: Substitute into the Binomial Theorem

Substituting these values into the binomial theorem gives us: `(2x)^3 + (C(3, 1) * (2x)^2 * 3) + (C(3, 2) * (2x) * 3^2) + 3^3`. Here, `(2x)^3` implies cube of 2x, `C(3, 1)` stands for the number of ways to select one element from three, similarly, `C(3, 2)` stands for the number of ways to select two elements from three.

Step 3: Compute the Binomial Coefficients

First, calculate the binomial coefficients. `C(3, 1)` is `3` and `C(3, 2)` is also `3`.

Step 4: Simplify

Now we simplify the entire expression, yielding `8x^3 + 36x^2 + 54x + 27`.