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

Question

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 \(z^2 - 16\)).