Question

Factor the polynomial if it is a perfect square​ trinomial, or state that the polynomial is prime.
x squared plus 20 xy plus 100 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 20 xy plus 100 y squaredequals
  
enter your response here ​(Factor completely.)
B.
The polynomial is prime.

Answer 1

To determine if the polynomial \( x^2 + 20xy + 100y^2 \) is a perfect square trinomial, we can check if it can be expressed in the form \( (ax + by)^2 \).

A perfect square trinomial has the following form:

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

In this polynomial:

- The first term \( x^2 \) suggests \( a = x \).
- The last term \( 100y^2 \) suggests \( b^2 = 100y^2 \), so \( b = 10y \).
- The middle term \( 20xy \) suggests \( 2ab = 20xy \).

Now plug in the values for \( a \) and \( b \):

\[
2(x)(10y) = 20xy
\]

Since the middle term \( 20xy \) matches, we can factor the polynomial as follows:

\[
x^2 + 20xy + 100y^2 = (x + 10y)^2
\]

Thus, the fully factored form is:

\[
(x + 10y)^2
\]

So, the correct choice is:

**A.** \( x^2 + 20xy + 100y^2 = (x + 10y)^2 \)