Which expression is equivalent to (z+4)(z−4)

Question

?(1 point)
Responses

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

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)

Answer 1

To simplify the expression \((z+4)(z−4)\), we can use the difference of squares formula, which states that \((a + b)(a - b) = a^2 - b^2\).

In this case:

• \(a = z\)
• \(b = 4\)

Therefore, we can apply the formula:

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

Now let’s compare this result with the given responses:

1. \(z^2 + 4z - 4z - 16\) simplifies to \(z^2 - 16\) (since \(4z - 4z = 0\)). This is equivalent to our result.
2. \(z^2 - 8z - 16\) is not equivalent.
3. \(2z^2 - z^2 - 8\) simplifies to \(z^2 - 8\), which is also not equivalent.
4. \(z(z) + 4z(z) - 4(4)\) expands to \(z^2 + 4z^2 - 16\) or \(5z^2 - 16\), which is not equivalent.

Thus, the correct equivalent expression is:

\(z^2 + 4z - 4z - 16\) (Option 1).