To factor the expression [tex]\(26 r^3 s + 52 r^5 - 39 r^2 s^4\)[/tex], we'll follow these steps:
1. Identify the greatest common factor (GCF) of all the terms.
2. Factor out the GCF from the expression.
Let's start with step 1:
### Step 1: Identify the GCF
- The coefficients are 26, 52, and 39. The GCF of these numbers is 13 because:
- The prime factorization of 26 is [tex]\(2 \times 13\)[/tex].
- The prime factorization of 52 is [tex]\(2^2 \times 13\)[/tex].
- The prime factorization of 39 is [tex]\(3 \times 13\)[/tex].
- Next, look at the variable parts:
- The terms are [tex]\(r^3 s\)[/tex], [tex]\(r^5\)[/tex], and [tex]\(r^2 s^4\)[/tex].
- The lowest power of [tex]\(r\)[/tex] is [tex]\(r^2\)[/tex].
- The lowest power of [tex]\(s\)[/tex] is [tex]\(s^0\)[/tex] (since [tex]\(r^5\)[/tex] does not have an [tex]\(s\)[/tex]).
So, the GCF of the variable parts is [tex]\(r^2\)[/tex].
### Step 2: Factor out the GCF
Now we factor out [tex]\(13 r^2\)[/tex] from the expression:
[tex]\[
26 r^3 s + 52 r^5 - 39 r^2 s^4 = 13 r^2 ( \frac{26 r^3 s}{13 r^2} + \frac{52 r^5}{13 r^2} + \frac{39 r^2 s^4}{13 r^2} )
\][/tex]
Simplify each term:
[tex]\[
= 13 r^2 \left( 2 r s + 4 r^3 - 3 s^4 \right)
\][/tex]
Therefore, the factored expression is:
[tex]\[
13 r^2 (2 r s + 4 r^3 - 3 s^4)
\][/tex]
From the given options, the correct choice is:
[tex]\[ \boxed{13 r^2 \left(2 r s + 4 r^3 - 3 s^4\right)} \][/tex]
