In Exercises 69–82, multiply using the rule for the product of the sum and difference of two terms. (y³ + 3)(y³ − 3)

Verified step by step guidance

1. Identify the expression as a product of the sum and difference of two terms: $(a+b)(a-b)$.
2. Recognize that $a=y^3$ and $b=3$ in the given expression $(y^3+3)(y^3-3)$.
3. Apply the formula for the product of the sum and difference of two terms: $(a+b)(a-b)=a^2-b^2$.
4. Substitute $a=y^3$ and $b=3$ into the formula: $(y^3{)}^2-3^2$.
5. Simplify the expression: $y^6-9$.

Key Concepts

Product of Sum and Difference

The product of the sum and difference of two terms follows the formula (a + b)(a - b) = a² - b². This identity simplifies the multiplication of expressions where one is the sum and the other is the difference of the same two terms, allowing for a straightforward calculation without expanding both expressions fully.

Polynomial Multiplication

Polynomial multiplication involves multiplying two polynomials to produce a new polynomial. In this case, we are dealing with polynomials of the form (y³ + 3) and (y³ - 3), where the multiplication can be simplified using the product of sum and difference rule, leading to a more efficient calculation.

Difference of Squares

The difference of squares is a specific case of the product of sum and difference, represented as a² - b². In the context of the given problem, y³ is treated as 'a' and 3 as 'b', allowing us to apply this concept to simplify the expression directly to (y³)² - (3)², which results in y^6 - 9.