Certainly! Let's analyze the given polynomial expressions one by one to determine which represents a sum of cubes. We'll expand each expression and identify its form.
1. For the expression [tex]\((6 - s)\left(s^2 + 6s + 36\right)\)[/tex]:
[tex]\[
(6-s)(s^2+6 s+36)
\][/tex]
Expanding this:
[tex]\[
= 6(s^2 + 6s + 36) - s(s^2 + 6s + 36)
\][/tex]
[tex]\[
= 6s^2 + 36s + 216 - s^3 - 6s^2 - 36s
\][/tex]
[tex]\[
= - s^3 + 216
\][/tex]
The result is:
[tex]\[
216 - s^3
\][/tex]
2. For the expression [tex]\((6 + s)\left(s^2 - 6s - 36\right)\)[/tex]:
[tex]\[
(6+s)(s^2-6 s-36)
\][/tex]
Expanding this:
[tex]\[
= 6(s^2 - 6s - 36) + s(s^2 - 6s - 36)
\][/tex]
[tex]\[
= 6s^2 - 36s - 216 + s^3 - 6s^2 - 36s
\][/tex]
[tex]\[
= s^3 - 72s - 216
\][/tex]
The result is:
[tex]\[
s^3 - 72s - 216
\][/tex]
3. For the expression [tex]\((6 + s)\left(s^2 - 6s + 36\right)\)[/tex]:
[tex]\[
(6+s)(s^2-6 s+36)
\][/tex]
Expanding this:
[tex]\[
= 6(s^2 - 6s + 36) + s(s^2 - 6s + 36)
\][/tex]
[tex]\[
= 6s^2 - 36s + 216 + s^3 - 6s^2 + 36s
\][/tex]
[tex]\[
= s^3 + 216
\][/tex]
The result is:
[tex]\[
s^3 + 216
\][/tex]
4. For the expression [tex]\((6 + s)\left(s^2 + 6s + 36\right)\)[/tex]:
[tex]\[
(6+s)(s^2+6 s+36)
\][/tex]
Expanding this:
[tex]\[
= 6(s^2 + 6s + 36) + s(s^2 + 6s + 36)
\][/tex]
[tex]\[
= 6s^2 + 36s + 216 + s^3 + 6s^2 + 36s
\][/tex]
[tex]\[
= s^3 + 12s^2 + 72s + 216
\][/tex]
The result is:
[tex]\[
(s + 6)(s^2 + 6s + 36)
\][/tex]
Now, a sum of cubes is in the form [tex]\(a^3 + b^3\)[/tex]. Looking at the forms obtained from the expansions above:
- [tex]\(216 - s^3\)[/tex] is of the form [tex]\(a^3 - b^3\)[/tex].
- [tex]\(s^3 - 72s - 216\)[/tex] does not match the sum of cubes form.
- [tex]\(s^3 + 216\)[/tex] is of the form [tex]\(a^3 + b^3\)[/tex] where [tex]\(a = s\)[/tex] and [tex]\(b = 6\)[/tex].
Therefore, the polynomial expression [tex]\((6 + s)(s^2 - 6s + 36)\)[/tex] represents a sum of cubes.
