Which polynomial expression represents a sum of cubes?

A. [tex]\((6-s)(s^2+6s+36)\)[/tex]

B. [tex]\((6+s)(s^2-6s-36)\)[/tex]

C. [tex]\((6+s)(s^2-6s+36)\)[/tex]

D. [tex]\((6+s)(s^2+6s+36)\)[/tex]



Answer :

To determine which polynomial expression represents a sum of cubes, we need to recall the formula for the sum of cubes, which is given by:

[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]

Let's analyze each given option to see if any of them match this formula.

1. Option 1: [tex]\((6-s)(s^2 + 6s + 36)\)[/tex]

Let's expand this expression:

[tex]\[ (6 - s)(s^2 + 6s + 36) = 6s^2 + 36s + 216 - s^3 - 6s^2 - 36s \][/tex]

Combining like terms:

[tex]\[ 0 - s^3 + 0 + 216 = -s^3 + 216 \][/tex]

This does not represent a sum of cubes.

2. Option 2: [tex]\((6 + s)(s^2 - 6s - 36)\)[/tex]

Let's expand this expression:

[tex]\[ (6 + s)(s^2 - 6s - 36) = 6s^2 - 36s - 216 + s^3 - 6s^2 - 36s \][/tex]

Combining like terms:

[tex]\[ s^3 + 0 - 72s - 216 \][/tex]

This does not represent a sum of cubes.

3. Option 3: [tex]\((6 + s)(s^2 - 6s + 36)\)[/tex]

Let's expand this expression:

[tex]\[ (6 + s)(s^2 - 6s + 36) = 6s^2 - 36s + 216 + s^3 - 6s^2 + 36s \][/tex]

Combining like terms:

[tex]\[ s^3 + 0 + 0 + 216 = s^3 + 216 \][/tex]

This represents [tex]\(s^3 + 6^3\)[/tex], which is indeed a sum of cubes. So, this option matches the sum of cubes formula when [tex]\(a = s\)[/tex] and [tex]\(b = 6\)[/tex].

4. Option 4: [tex]\((6 + s)(s^2 + 6s + 36)\)[/tex]

Let's expand this expression:

[tex]\[ (6 + s)(s^2 + 6s + 36) = 6s^2 + 36s + 216 + s^3 + 6s^2 + 36s \][/tex]

Combining like terms:

[tex]\[ s^3 + 12s^2 + 72s + 216 \][/tex]

This does not represent a sum of cubes.

Given our analysis, the polynomial expression that represents a sum of cubes is:

[tex]\[ \boxed{(6 + s)(s^2 - 6s + 36)} \][/tex]