Let's analyze the expression [tex]\( 27m^9 - 8n^{15}p^{21} \)[/tex] to identify its factors.
### Step 1: Identify the general form of the factors
The expression has the form of a difference of cubes:
[tex]\[ 27m^9 - 8n^{15}p^{21} \][/tex]
We can rewrite [tex]\( 27m^9 \)[/tex] as [tex]\( (3m^3)^3 \)[/tex] and [tex]\( 8n^{15}p^{21} \)[/tex] as [tex]\( (2n^5p^7)^3 \)[/tex]. So, this expression can be thought of as:
[tex]\[ (3m^3)^3 - (2n^5p^7)^3 \][/tex]
### Step 2: Apply the difference of cubes formula
The difference of cubes is factored using the formula:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
Substitute [tex]\( a = 3m^3 \)[/tex] and [tex]\( b = 2n^5p^7 \)[/tex]:
[tex]\[ (3m^3)^3 - (2n^5p^7)^3 = (3m^3 - 2n^5p^7)((3m^3)^2 + (3m^3)(2n^5p^7) + (2n^5p^7)^2) \][/tex]
### Step 3: Simplify each term in the factors
First factor:
[tex]\[ (3m^3 - 2n^5p^7) \][/tex]
Second factor:
[tex]\[ ((3m^3)^2 + (3m^3)(2n^5p^7) + (2n^5p^7)^2) \][/tex]
Calculate:
[tex]\[
(3m^3)^2 = 9m^6
\][/tex]
[tex]\[
(3m^3)(2n^5p^7) = 6m^3 n^5 p^7
\][/tex]
[tex]\[
(2n^5p^7)^2 = 4n^{10}p^{14}
\][/tex]
So, the second factor becomes:
[tex]\[ 9m^6 + 6m^3 n^5 p^7 + 4n^{10} p^{14} \][/tex]
### Step 4: Combine the results
The factored form of the expression [tex]\( 27m^9 - 8n^{15}p^{21} \)[/tex] is:
[tex]\[ (3m^3 - 2n^5p^7)(9m^6 + 6m^3 n^5 p^7 + 4n^{10} p^{14}) \][/tex]
### Step 5: Match the factors with provided options
We compare each option to see which one matches one of the factors from our result.
- [tex]\(\left(9 m^9 + 6m^3 n^4 p^7 + 4n^{25} p^{49}\right)\)[/tex] does not match.
- [tex]\(\left(3m^3 + 2n^5 p^7 \right)\)[/tex] does not match.
- [tex]\(\left(9 m^9 - 6 m^3 n^4 p^7 + 4 n^{10} p^{14}\right)\)[/tex] does not match.
- [tex]\(\left(3 m^3 - 2 n^5 p^7 \right)\)[/tex] matches the first factor.
Thus, the correct answer is:
[tex]\[ \boxed{3m^3 - 2n^5p^7} \][/tex]
This corresponds to option (4). Hence, the factor [tex]\(3m^3 - 2n^5 p^7\)[/tex] is one of the factors of the given expression.
