Question

Which of the following properly uses a polynomial identity to detect if  22k−1  is prime?(1 point)Responses22k−1=(2k+1)(2k−1), so 22k−1 is not prime.2 superscript 2 k baseline minus 1 equals left parenthesis 2 superscript k baseline plus 1 right parenthesis left parenthesis 2 superscript k baseline minus 1 right parenthesis , so  2 superscript 2 k baseline minus 1  is not prime.22k−1=(2k+1)(2k−1), so 22k−1 is prime.  2 superscript 2 k baseline minus 1 equals left parenthesis 2 superscript k baseline plus 1 right parenthesis left parenthesis 2 superscript k baseline minus 1 right parenthesis , so  2 superscript 2 k baseline minus 1  is prime.22k−1=(22+1)(2k−1), so 22k−1 is not prime.    2 superscript 2 k baseline minus 1 equals left parenthesis 2 superscript 2 baseline plus 1 right parenthesis left parenthesis 2 superscript k baseline minus 1 right parenthesis , so  2 superscript 2 k baseline minus 1  is not prime.22k−1=(22+1)(2k−1), so 22k−1 is prime.

Answer 1

To determine if \( 2^{2k} - 1 \) is prime using a polynomial identity, we can use the identity:

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

In our case, we can let \( a = 2^k \) and \( b = 1 \). Therefore:

\[
2^{2k} - 1 = (2^k - 1)(2^k + 1)
\]

This shows that \( 2^{2k} - 1 \) can be factored into two non-trivial factors: \( (2^k - 1) \) and \( (2^k + 1) \). For \( k \geq 1 \), both factors are greater than 1, which means \( 2^{2k} - 1 \) is not prime.

Based on the provided responses, the correct response is:

\[
2^{2k} - 1 = (2^k + 1)(2^k - 1), \text{ so } 2^{2k} - 1 \text{ is not prime.}
\]

Therefore, the correct option is:

\[
2^{2k} - 1 = (2^k + 1)(2^k - 1), \text{ so } 2^{2k} - 1 \text{ is not prime.}
\]