Answer :
We start by recognizing that the given polynomial
[tex]$$
x^3 + 64
$$[/tex]
can be written as
[tex]$$
x^3 + 4^3,
$$[/tex]
because [tex]$64 = 4^3$[/tex]. This shows that the polynomial is a sum of cubes.
The sum of cubes has a general factorization formula given by
[tex]$$
a^3 + b^3 = (a + b)(a^2 - ab + b^2).
$$[/tex]
Here, we let [tex]$a = x$[/tex] and [tex]$b = 4$[/tex]. Substituting these values into the formula, we get
[tex]$$
x^3 + 4^3 = (x + 4)(x^2 - 4x + 16).
$$[/tex]
Thus, we can conclude that [tex]$x^3 + 64$[/tex] is an example of a sum of cubes, and its complete factorization is
[tex]$$
(x + 4)(x^2 - 4x + 16).
$$[/tex]
This detailed solution shows that the polynomial belongs to the "sum of cubes" category and confirms its factorization.
[tex]$$
x^3 + 64
$$[/tex]
can be written as
[tex]$$
x^3 + 4^3,
$$[/tex]
because [tex]$64 = 4^3$[/tex]. This shows that the polynomial is a sum of cubes.
The sum of cubes has a general factorization formula given by
[tex]$$
a^3 + b^3 = (a + b)(a^2 - ab + b^2).
$$[/tex]
Here, we let [tex]$a = x$[/tex] and [tex]$b = 4$[/tex]. Substituting these values into the formula, we get
[tex]$$
x^3 + 4^3 = (x + 4)(x^2 - 4x + 16).
$$[/tex]
Thus, we can conclude that [tex]$x^3 + 64$[/tex] is an example of a sum of cubes, and its complete factorization is
[tex]$$
(x + 4)(x^2 - 4x + 16).
$$[/tex]
This detailed solution shows that the polynomial belongs to the "sum of cubes" category and confirms its factorization.
