Question

Factor the polynomial if it is a perfect square​ trinomial, or state that the polynomial is prime.
x squared plus 24 xy plus 144 y squared
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
x squared plus 24 xy plus 144 y squaredequals
  
enter your response here ​(Factor completely.)
B.
The polynomial is prime.

Answer 1

To determine if the polynomial \( x^2 + 24xy + 144y^2 \) is a perfect square trinomial, we can look for factors that take the form \( (ax + by)^2 \).

A perfect square trinomial follows the pattern:

\[
(a + b)^2 = a^2 + 2ab + b^2
\]

Now, let's analyze the given polynomial:

1. **Identify \( a^2 \)**: The first term \( x^2 \) suggests that \( a = x \).

2. **Identify \( b^2 \)**: The last term \( 144y^2 \) suggests that \( b = 12y \) (since \( (12y)^2 = 144y^2 \)).

3. **Check the middle term**: The middle term \( 24xy \) should then fit the \( 2ab \) form:
\[
2ab = 2(x)(12y) = 24xy
\]

Since all the terms satisfy the conditions of a perfect square trinomial, we can write:

\[
x^2 + 24xy + 144y^2 = (x + 12y)^2
\]

Thus, the factorization is:

\[
\boxed{(x + 12y)^2}
\]