Which of the following is a factor of x15 − 64?

To determine which of the given options is a factor of \( x^{15} - 64 \), we identify that \( x^{15} - 64 \) can be recognized as a difference of squares because it can be rewritten as:

\[
x^{15} - 64 = x^{15} - 8^2
\]

Using the difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \), we can express this as:

\[
(x^{\frac{15}{2}} - 8)(x^{\frac{15}{2}} + 8)
\]

However, since \( x^{15} \) has a non-integer exponent in the context of that factorization, we can factor it in a more straightforward way as follows. The expression can also be modeled using the fact that \( 64 \) is \( 2^6 \):

To find the factors, we will need to determine if any of the proposed options can be simplified to give \( x^{15} - 64 \).

Let’s evaluate each option one at a time:

1. **Evaluate \( x^5 - 4 \)**:
\[
x^5 - 4 = x^5 - 2^2 \quad (\textit{difference of squares})
\]
It can be factored as:
\[
(x^{5/2} - 2)(x^{5/2} + 2)
\]
However, this isn't a direct factor of \( x^{15} - 64 \).

2. **Evaluate \( x^{10} - 4x^5 + 16 \)**:
This is a quadratic in terms of \( x^5 \):
\[
= (x^5)^2 - 4(x^5) + 16
\]
Using the quadratic formula:
\[
x^5 = \frac{4 \pm \sqrt{(-4)^2 - 4 \times 1 \times 16}}{2 \times 1}
\]
The discriminant \( (-4)^2 - 4 \cdot 1 \cdot 16 = 16 - 64 = -48 \) indicates that there are no real roots. Therefore, it’s unlikely to be a simple factor.

3. **Evaluate \( x^{10} - 4 \)**:
\[
x^{10} - 4 = x^{10} - 2^2 \quad (\textit{difference of squares})
\]
Factoring yields:
\[
(x^5 - 2)(x^5 + 2)
\]
\( x^{15} - 64 \) can indeed be expressed through \( (x^5 - 2)^n \) type structure because raising to 2 will yield similar terms.

4. **Evaluate \( x^{15} + 8x + 16 \)**:
This doesn’t obviously relate to \( x^{15} - 64 \), nor does it fit structured factorization for this polynomial.

The one with viable algebraic manipulation leading to a consistent outcome of polynomial degree and roots remaining is:

The relevant factor of \( x^{15} - 64 \) amongst the given options is:

**\( x^{10} - 4 \)**.