Question

suppose you want to factor the expression ( x^2 + 2xn + n^2 ). given that ( n > 0 ), what are the factors? explain. select the correct answer below and, if necessary, fill in the answer box to complete your choice. a. the value of ( n ) can be any positive integer resulting in the distinct factors (use a comma to separate answers as needed.) b. the value of ( n ) can be any prime integer resulting in the distinct factors (use a comma to separate answers as needed.) c. the value of ( n ) can be any positive integer resulting in the same factor d. the value of ( n ) can be any prime integer resulting in the same factor e. the expression cannot be factored for the given values of ( n ).

Explanation:

Step1: Recognize perfect square trinomial

Recall the identity: $a^2 + 2ab + b^2 = (a+b)^2$

Step2: Match to given expression

For $x^2 + 2xn + n^2$, set $a=x$, $b=n$.
Expression becomes $(x+n)^2$

Step3: Analyze factor form

The factored form is a single repeated factor, regardless of positive integer $n$.

Answer:

C. The value of n can be any positive integer resulting in the same factor $(x+n)$.