To determine which polynomial expression represents a sum of cubes, we can use the standard sum of cubes formula:
[tex]\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\][/tex]
We need to match this form with one of the given polynomial expressions. Let us examine them one by one:
1. [tex]\((6-s)\left(s^2 + 6s + 36\right)\)[/tex]
2. [tex]\((6+s)\left(s^2 - 6s - 36\right)\)[/tex]
3. [tex]\((6+s)\left(s^2 - 6s + 36\right)\)[/tex]
4. [tex]\((6+s)\left(s^2 + 6s + 36\right)\)[/tex]
We need to find expressions in the form of [tex]\( (a + b)(a^2 - ab + b^2) \)[/tex] where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] can be identified easily.
Let's analyze each option to see if they fit this format:
1. [tex]\((6-s)(s^2 + 6s + 36)\)[/tex]
- Here, the term [tex]\( (6-s) \)[/tex] does not match [tex]\( (a + b) \)[/tex] since it has a minus sign instead of a plus sign.
2. [tex]\((6+s)(s^2 - 6s - 36)\)[/tex]
- Here, even though the linear term [tex]\( (6+s) \)[/tex] might match [tex]\( (a + b) \)[/tex], the quadratic term [tex]\( s^2 - 6s - 36 \)[/tex] does not match [tex]\( a^2 - ab + b^2 \)[/tex]. The [tex]\(-36\)[/tex] should be positive.
3. [tex]\((6+s)(s^2 - 6s + 36)\)[/tex]
- This resembles [tex]\( (a + b) \)[/tex] as [tex]\( (a + b) \)[/tex], and [tex]\( s^2 - 6s + 36 \)[/tex] conforms to [tex]\( a^2 - ab + b^2 \)[/tex], so it is a valid candidate.
4. [tex]\((6+s)(s^2 + 6s + 36)\)[/tex]
- This option has the right structure for [tex]\( (a + b) \)[/tex] as [tex]\( (a + b) \)[/tex], but the term [tex]\( s^2 + 6s + 36 \)[/tex] does not match [tex]\( a^2 - ab + b^2 \)[/tex] since the sign in the middle term [tex]\(6s\)[/tex] should be negative.
By eliminating other options based on the mismatches, we conclude that the polynomial expression that represents a sum of cubes is:
[tex]\[
(6+s)(s^2-6s+36)
\][/tex]
So, the correct answer is:
[tex]\[
(6+s)\left(s^2 - 6s + 36\right)
\][/tex]
The fourth polynomial expression represents the sum of cubes.
