In Exercises 67–82, find each product. (7xy^2−10y)(7xy^2+10y)

Verified step by step guidance

1. Recognize that the expression is a product of two binomials: \((7xy^2 - 10y)\) and \((7xy^2 + 10y)\).
2. Notice that this is a difference of squares pattern, which can be expressed as \((a - b)(a + b) = a^2 - b^2\).
3. Identify \(a = 7xy^2\) and \(b = 10y\) in the expression.
4. Apply the difference of squares formula: \((7xy^2)^2 - (10y)^2\).
5. Calculate each square: \((7xy^2)^2 = 49x^2y^4\) and \((10y)^2 = 100y^2\).

Key Concepts

Factoring and the Difference of Squares

The expression given is in the form of a difference of squares, which follows the formula a^2 - b^2 = (a - b)(a + b). In this case, a is 7xy^2 and b is 10y. Recognizing this pattern allows for simplification and efficient calculation of the product.

Distributive Property

The distributive property states that a(b + c) = ab + ac. This property is essential for expanding expressions and multiplying polynomials. In the context of the given problem, applying the distributive property will help in calculating the product of the two binomials.

Combining Like Terms

When multiplying polynomials, it is crucial to combine like terms to simplify the expression. Like terms are terms that have the same variable raised to the same power. After applying the distributive property, identifying and combining these terms will yield the final simplified product.