Answer :

The polynomial [tex]\( x^3 + 64 \)[/tex] is an example of a "sum of cubes."

Here's how you can identify and factor it:

1. Recognize the Form:
The expression [tex]\( x^3 + 64 \)[/tex] fits the structure of a sum of cubes, which is generally represented as [tex]\( a^3 + b^3 \)[/tex].

2. Identify the Terms:
In the polynomial [tex]\( x^3 + 64 \)[/tex]:
- [tex]\( a^3 = x^3 \)[/tex], so [tex]\( a = x \)[/tex].
- [tex]\( b^3 = 64 \)[/tex]. To find [tex]\( b \)[/tex], note that [tex]\( 64 = 4^3 \)[/tex], so [tex]\( b = 4 \)[/tex].

3. Use the Sum of Cubes Formula:
The sum of cubes can be factored using the formula:
[tex]\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\][/tex]
Substituting [tex]\( a = x \)[/tex] and [tex]\( b = 4 \)[/tex] into this formula gives:
[tex]\[
x^3 + 64 = (x + 4)(x^2 - 4x + 16)
\][/tex]

4. Conclusion:
Therefore, [tex]\( x^3 + 64 \)[/tex] is a sum of cubes, and it can be factored as [tex]\( (x + 4)(x^2 - 4x + 16) \)[/tex].

This approach shows that the polynomial you started with is indeed a perfect example of a sum of cubes and how it can be factorized.