In perfect square trinomials, is x^2 + 6xy + 36y^2 prime?

The expression \( x^2 + 6xy + 36y^2 \) is not a prime polynomial, but it is not a perfect square trinomial either.

### Explanation:
To determine if the trinomial is a perfect square, we check if it can be written in the form \( (ax + by)^2 \).

1. **Perfect Square Check**:
   A perfect square trinomial takes the form:
   \[
   a^2 + 2ab + b^2 = (a + b)^2
   \]
   Compare \( x^2 + 6xy + 36y^2 \) with \( a^2 + 2ab + b^2 \).

   Here, \( a^2 = x^2 \), so \( a = x \).

   Next, compare the middle term \( 2ab = 6xy \). Substituting \( a = x \):
   \[
   2x \cdot b = 6xy \implies b = 3y
   \]
   
   Now, let's check if \( b^2 = (3y)^2 = 9y^2 \) matches the third term \( 36y^2 \). However, \( 9y^2 \) does not equal \( 36y^2 \), so the expression is not a perfect square trinomial.

2. **Factorization**:
   Let's see if the expression can be factored differently:
   \[
   x^2 + 6xy + 36y^2
   \]
   The expression doesn't factor into linear factors easily, but by completing the square, we find that it does not yield simple rational factors either. This trinomial does not factor nicely over the integers or simple expressions.

Therefore, the expression is **not prime** in the traditional sense because it’s possible to factor it over complex numbers, but it's also **not** a perfect square trinomial.

Would you like more details or have any questions?

Here are some related questions:

1. How can you factor trinomials that are not perfect squares?
2. What are some examples of trinomials that are neither prime nor perfect squares?
3. How can you determine if a polynomial is prime?
4. What is the process of completing the square, and how does it help in factoring?
5. How does the discriminant help in identifying the factorability of quadratic polynomials?

**Tip:** Always verify each term carefully when checking if a trinomial is a perfect square or if it can be factored into simpler expressions.

Explore whether the expression x^2 + 6xy + 36y^2 is a perfect square trinomial or a prime polynomial. Understand the criteria for perfect square trinomials and the factorization process. Detailed explanation and analysis suitable for advanced high school students.

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