Find each product. (2x+3)(2x-3)(4x²-9)

Verified step by step guidance

1. Recognize that the expression is a product of three factors: \((2x+3)\), \((2x-3)\), and \((4x^2-9)\).
2. First, multiply the two binomials \((2x+3)\) and \((2x-3)\) using the difference of squares formula: \((a+b)(a-b) = a^2 - b^2\). Here, \(a = 2x\) and \(b = 3\), so the product is \( (2x)^2 - 3^2 \).
3. Calculate the squares: \((2x)^2 = 4x^2\) and \(3^2 = 9\), so the product of the first two binomials is \(4x^2 - 9\).
4. Now, notice that the product of the first two binomials is exactly the third factor, \((4x^2 - 9)\), so the entire expression becomes \((4x^2 - 9)(4x^2 - 9)\).
5. Finally, multiply \((4x^2 - 9)\) by itself, which is a perfect square trinomial: \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = 4x^2\) and \(b = 9\), so expand accordingly.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying each term in one polynomial by every term in the other polynomial(s) and then combining like terms. This process is essential for expanding expressions such as (2x+3)(2x-3)(4x^2-9).

Difference of Squares

The difference of squares is a special product formula: a^2 - b^2 = (a + b)(a - b). Recognizing this pattern helps simplify expressions like (2x+3)(2x-3), which equals 4x^2 - 9, making multiplication more efficient.

Combining Like Terms

After multiplying polynomials, combining like terms means adding or subtracting terms with the same variable and exponent. This step simplifies the expression into its standard polynomial form, making it easier to interpret or use in further calculations.