Question

Which expression is equivalent to (z+4)(z−4) ?(1 point) Responses 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 z squared plus 4 z minus 4 z minus 16 2z2−z2−8 2 z squared minus z squared minus 8 z2−8z−16

Answer 1

To find an expression equivalent to \((z+4)(z−4)\), we can apply the difference of squares formula, which states that:

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

In this case, \(a = z\) and \(b = 4\). Thus, we have:

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

So the equivalent expression is:

\[
z^2 - 16
\]

Now, reviewing the options you've provided, the correct option is not explicitly listed, but it can be inferred from the expressions given. The choice that explicitly outlines the simplification as \(z^2 - 16\) can be interpreted as equivalent.

### To interpret the provided options:

1. **z(z)+4z(z)−4(4)** - This does not equal our target.
2. **z(z)+4z(z)−4(4)** - This is the same as the above. Incorrect.
3. **z^2 + 4z − 4z − 16** - This simplifies to **z^2 - 16**. This is equivalent.
4. **2z^2 − z^2 − 8** - This simplifies to **z^2 - 8**, which is not equivalent.
5. **z^2 − 8z − 16** - This does not equal our target.

Thus, the expression **z² + 4z − 4z − 16** correctly simplifies to **z² - 16** and is equivalent to \((z+4)(z−4)\).