Question

Which expression is equivalent to \(\left(z+4\right)\left(z-4\right)\)?(1 point)
Responses

\(z^2-8z-16\)
z squared minus 8 z minus 16

\(z(z)+4z(z)-4(4)\)
z left parenthesis z right parenthesis plus 4 z left parenthesis z right parenthesis minus 4 left parenthesis 4 right parenthesis

\(z^2+4z-4z-16\)
z squared plus 4 z minus 4 z minus 16

\(2z^2-z^2-8\)

Answer 1

To find an equivalent expression for \(\left(z+4\right)\left(z-4\right)\), we can apply the difference of squares formula, which states:

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

In this case, we can identify \(a = z\) and \(b = 4\), so we have:

\[
(z+4)(z-4) = z^2 - 4^2 = z^2 - 16
\]

Now let's compare this with the given options:

1. \(z^2-8z-16\) : This does not match.
2. \(z(z)+4z(z)-4(4)\) : This is not in standard form and does not match.
3. \(z^2+4z-4z-16\): This simplifies to \(z^2 - 16\), which matches our result.
4. \(2z^2-z^2-8\): This simplifies to \(z^2 - 8\), which does not match.

Therefore, the expression equivalent to \(\left(z+4\right)\left(z-4\right)\) is:

\[
\boxed{z^2-16}
\]

Only option 3 is equivalent upon simplification, even though it is not directly stated as \(z^2 - 16\).