Find each product. See Examples 5 and 6. (4x^2-5y0(4x^2+5y)

Verified step by step guidance

1. Identify the expression to be multiplied: \((4x^2 - 5y)(4x^2 + 5y)\).
2. Recognize that this is a difference of squares pattern: \((a - b)(a + b) = a^2 - b^2\).
3. Assign \(a = 4x^2\) and \(b = 5y\).
4. Apply the difference of squares formula: \((4x^2)^2 - (5y)^2\).
5. Simplify each term: \((4x^2)^2 = 16x^4\) and \((5y)^2 = 25y^2\).

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves distributing each term in one polynomial to every term in another polynomial. This process is often referred to as the FOIL method for binomials, which stands for First, Outside, Inside, Last. Understanding how to combine like terms after distribution is crucial for simplifying the resulting expression.

Like Terms

Like terms are terms that have the same variable raised to the same power. For example, in the expression 4x^2 and -5x^2, both terms are like terms because they share the same variable x raised to the second power. Identifying and combining like terms is essential for simplifying polynomial expressions after multiplication.

Zero Exponent Rule

The zero exponent rule states that any non-zero number raised to the power of zero equals one. This rule is important when dealing with terms that may have a variable or constant raised to the zero power, as it simplifies calculations and helps in understanding the structure of polynomial expressions.