Given the polynomial:
[tex]\[ 3 p^2 - 15 p q + 20 q^2 \][/tex]
We want to factorize it and determine which of the given options correctly represents its factorization.
### Step-by-Step Solution:
1. Identify the polynomial given:
[tex]\[ 3 p^2 - 15 p q + 20 q^2 \][/tex]
2. Examine the factorization of the polynomial:
We need to explore possible factor combinations. The form of the factors is usually based on the coefficients and the terms present.
3. Test each of the given options to see if they match the original polynomial after expansion:
Option A: [tex]\( (3p - 5q)(p - 10q) \)[/tex]
- Expand this product:
[tex]\[
(3p - 5q)(p - 10q) = 3p \cdot p + 3p \cdot (-10q) + (-5q) \cdot p + (-5q) \cdot (-10q)
\][/tex]
[tex]\[
= 3p^2 - 30pq - 5pq + 50q^2
\][/tex]
[tex]\[
= 3p^2 - 35pq + 50q^2
\][/tex]
- This does not match the original polynomial.
Option B: [tex]\( (3p - 5q)(p + 5q) \)[/tex]
- Expand this product:
[tex]\[
(3p - 5q)(p + 5q) = 3p \cdot p + 3p \cdot 5q + (-5q) \cdot p + (-5q) \cdot 5q
\][/tex]
[tex]\[
= 3p^2 + 15pq - 5pq - 25q^2
\][/tex]
[tex]\[
= 3p^2 + 10pq - 25q^2
\][/tex]
- This also does not match the original polynomial.
Option C: [tex]\( (3p - 5q)(2p^2 + 5q) \)[/tex]
- Expand this product:
[tex]\[
(3p - 5q)(2p^2 + 5q) = 3p \cdot 2p^2 + 3p \cdot 5q - 5q \cdot 2p^2 - 5q \cdot 5q
\][/tex]
[tex]\[
= 6p^3 + 15pq - 10pq^2 - 25q^2
\][/tex]
- This also does not match the original polynomial.
4. After testing the given options, none of them correctly factorizes the polynomial to match the original expression.
### Conclusion:
- The correct choice is:
D. The polynomial is irreducible.
Therefore, the polynomial [tex]\( 3 p^2 - 15 p q + 20 q^2 \)[/tex] cannot be factorized into simpler rational polynomials, and the correct answer is:
[tex]\[ \boxed{D} \][/tex]
