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

Question

z^2−8z−16
2z^2−z^2−8
z(z)+4z(z)−4(4)
z^2+4z−4z−16

Answer 1

The expression \((z+4)(z-4)\) can be simplified using the difference of squares formula, which states that:

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

In this case, \(a = z\) and \(b = 4\). Therefore:

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

Now let's compare this result with the given options:

1. \(z^2 - 8z - 16\)
2. \(2z^2 - z^2 - 8\)
3. \(z(z) + 4z(z) - 4(4)\)
4. \(z^2 + 4z - 4z - 16\)

The expression equivalent to \(z^2 - 16\) is not directly listed, but let's analyze each option:

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

Thus, the expression that is equivalent to \((z+4)(z-4)\) is:

\(z^2 - 16\) from option 4.