Find each product. See Example 5. (4x² - 5y) (4x² + 5y)

Verified step by step guidance

1. Recognize that the expression is a product of two binomials in the form \((a - b)(a + b)\), where \(a = 4x^{2}\) and \(b = 5y\).
2. Recall the difference of squares formula: \((a - b)(a + b) = a^{2} - b^{2}\).
3. Apply the formula by squaring each term: calculate \(a^{2} = (4x^{2})^{2}\) and \(b^{2} = (5y)^{2}\).
4. Write the expression as \(a^{2} - b^{2}\), which becomes \((4x^{2})^{2} - (5y)^{2}\).
5. Simplify the squares by applying the exponent rules: \((4x^{2})^{2} = 4^{2} imes (x^{2})^{2}\) and \((5y)^{2} = 5^{2} imes y^{2}\).

Key Concepts

Difference of Squares

The difference of squares is a special product formula: (a - b)(a + b) = a² - b². It simplifies the multiplication of two binomials that are conjugates by subtracting the square of the second term from the square of the first.

Polynomial Multiplication

Polynomial multiplication involves multiplying each term in the first polynomial by each term in the second polynomial and then combining like terms. This process is essential for expanding expressions and simplifying products.

Exponents and Like Terms

Understanding exponents is crucial when squaring terms like 4x², which becomes (4x²)² = 16x⁴. Combining like terms means adding or subtracting terms with the same variable and exponent to simplify the final expression.