Final answer:
The difference of two squares 16x⁴ - 81 is factored by taking the square root of each term and applying the formula a² - b² = (a + b)(a - b), resulting in (4x² + 9)(2x + 3)(2x - 3).
Step-by-step explanation:
Finding the Rational Factors of the Difference of Two Squares
The expression 16x⁴ - 81 represents a difference of two squares since 16x⁴ is the square of 4x² and 81 is the square of 9. To factor the difference of two squares, we take the square root of each term and use the formula a² - b² = (a + b)(a - b). In this case, the square roots are 4x² and 9, giving us:
(4x² + 9)(4x² - 9)
Note that the term 4x² - 9 is itself a difference of squares, which can be further factored as:
(4x² + 9)(2x + 3)(2x - 3)
This final expression shows the factors of the original expression, which are understandable and easy to work with. This kind of factorization is useful in a variety of algebraic manipulations and problems.