Question

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

z2−8z−16
z squared minus 8 z minus 16

z2+4z−4z−16
z squared plus 4 z minus 4 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

Answer 1

To find an equivalent expression for \((z+4)(z−4)\), we can recognize that this is a difference of squares. The formula for the difference of squares is:

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

In this case, we have:

- \(a = z\)
- \(b = 4\)

Therefore, we can rewrite the expression as:

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

Now, let's evaluate the provided options to find the one that matches \(z^2 - 16\):

1. **\(z^2 - 8z - 16\)**: This does not match.
2. **\(z^2 + 4z - 4z - 16\)**: This simplifies to \(z^2 - 16\), which does match.
3. **\(2z^2 - z^2 - 8\)**: This simplifies to \(z^2 - 8\), which does not match.
4. **\(z(z) + 4z(z) - 4(4)\)**: This expression does not simplify to \(z^2 - 16\).

The correct response is:

**z² + 4z − 4z − 16** (This simplifies to \(z^2 - 16\)).