Problem 53 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The expanded form of \((2x+3)^3\) is \(8x^3 + 36x^2 + 54x + 27\).

Step 1: Express the Cubic Function as a Binomial Expression

We can rewrite the given expression \((2x+3)^3\) as a binomial expression to the third power. This makes it easier to apply the binomial theorem.

Step 2: Apply the Binomial Theorem

Next, we use the binomial theorem, which states that \((a+b)^n = \sum_{k=0}^n {n \choose k} a^{n-k} b^k\). In our case, \(a = 2x\), \(b = 3\), and \(n = 3\), so: \((2x+3)^3 = {3 \choose 0} * (2x)^3 * 3^0 + {3 \choose 1} * (2x)^2 * 3^1 + {3 \choose 2} * (2x)^1 * 3^2 + {3 \choose 3} * (2x)^0 * 3^3\)

Step 3: Calculate the Coefficients

To simplify, we calculate each coefficient:\({3 \choose 0} = 1, {3 \choose 1} = 3, {3 \choose 2} = 3, {3 \choose 3} = 1\)and \(3^0 = 1, 3^1 = 3, 3^2 = 9, 3^3 = 27\)and \((2x)^0 = 1, (2x)^1 = 2x, (2x)^2 = 4x^2, (2x)^3 = 8x^3\)We then substitute these values back into the equation from the previous step.

Step 4: Simplify the Expression

After substitifying, the expression becomes:\(1*(8x^3)*1 + 3*(4x^2)*3 + 3*(2x)*9 + 1*1*27 = 8x^3 + 36x^2 + 54x + 27\)