Which expression is equivalent to (z+4)(z−4)?(1 point)

Question

Responses

z2−8z−16
z squared minus 8 z minus 16

2z2−z2−8
2 z squared minus z squared minus 8

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

z2+4z−4z−16

Answer 1

To determine which expression is equivalent to \((z+4)(z−4)\), we can apply the difference of squares formula. The expression is in the form of \((a+b)(a-b)\), which simplifies to \(a^2 - b^2\).

Here, \(a = z\) and \(b = 4\).

Thus, we have: \[ (z+4)(z−4) = z^2 - 4^2 = z^2 - 16. \]

Now, let's examine the given responses:

1. \(z^2 - 8z - 16\) - This is not equivalent because it has a linear term.
2. \(2z^2 - z^2 - 8\) - This simplifies to \(z^2 - 8\), which is not equivalent.
3. \(z(z) + 4z(z) - 4(4)\) - This expression does not simplify correctly to match \(z^2 - 16\).
4. \(z^2 + 4z - 4z - 16\) - This simplifies to \(z^2 - 16\) because the \(4z\) and \(-4z\) cancel each other out.

The correct equivalent expression is:

\(z^2 - 16\) (which corresponds to the response option that simplifies to \(z^2 - 16\)).