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

Question

Responses

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

z2−8z−16
z squared minus 8 z minus 16

z2+4z−4z−16

Answer 1

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

In this case, \(a = z\) and \(b = 4\). Thus, we can apply the formula:

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

Now, let's analyze the given response options to see which one matches \(z^2 - 16\):

1. 2z² - z² - 8: This simplifies to \(z² - 8\), which is not equivalent.

2. z(z) + 4z(z) - 4(4): This expands to \(z² + 4z² - 16 = 5z² - 16\), which is not equivalent.

3. z² - 8z - 16: This has a middle term \(-8z\), which is not equivalent.

4. z² + 4z - 4z - 16: This simplifies to \(z² - 16\), which is equivalent to our original expression.

Therefore, the correct response is:

z² + 4z - 4z - 16.