Final answer:
The correct factor of x¹⁵ - 64 is x⁵ - 4, derived from recognizing the expression as a difference of two cubes. So the correct factor is x⁵ - 4, which corresponds to option A.
Step-by-step explanation:
The student's question asks which of the following options is a factor of x¹⁵-64:
- x⁵-4
- x¹⁰-4x⁵+16
- x¹⁰-4
- x¹⁵+8x+16
To find the correct factor, we can use the difference of two squares, which implies that a² - b² = (a + b)(a - b). In the case of x¹⁵ - 64, we can rewrite 64 as 4³ and notice that x¹⁵ is the 3rd power of x⁵. Thus, we have (x⁵)³ - (4)³, which is a difference of two cubes. The factorization of a difference of two cubes is a³ - b³ = (a - b)(a² + ab + b²). In this case, that translates to (x⁵ - 4)(x¹⁰ + 4x⁵ + 16). So the correct factor is x⁵ - 4, which corresponds to option A.