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Sagot :
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.
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.
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