Final answer:
The correct expansion of the expression (16x⁴ + 1)(8x³ + 1)(4x² + 1)(2x + 1)(2x - 1) is 32x¹⁰ - 1, which is option a) 32x¹⁰ - 16x⁸ + 8x⁶ - 4x⁴ + 2x² - 1. This is achieved by recognizing the pattern of the difference of squares in the expression.
Step-by-step explanation:
The correct expansion of the expression (16x⁴ + 1)(8x³ + 1)(4x² + 1)(2x + 1)(2x - 1) is 32x¹⁰ - 1, which is option a) 32x¹⁰ - 16x⁸ + 8x⁶ - 4x⁴ + 2x² - 1. This is achieved by recognizing the pattern of the difference of squares in the expression. The student is asked to expand the expression (16x⁴ + 1)(8x³ + 1)(4x² + 1)(2x + 1)(2x - 1). This type of problem in mathematics involves using algebraic manipulation and the application of the binomial theorem to expand expressions containing powers of binomials. In this particular case, there is a shortcut.
The expression has the form of a difference of squares since (2x + 1)(2x - 1) = (2x)² - 1². Applying this to the given expression yields (16x⁴ + 1)·(8x³ + 1)·(4x² + 1)·((2x)² - 1), which simplifies to (16x⁴ + 1)·(8x³ + 1)·(4x² + 1)·(4x² - 1). The pattern continues as we see that (4x² + 1)(4x² - 1) is also a difference of squares, leading to further simplification until we end up with the final expression, 32x¹⁰ - 1. Therefore, option a) 32x¹⁰ - 16x⁸ + 8x⁶ - 4x⁴ + 2x² - 1 is correct.