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

2z2−z2−8
2 z squared minus z squared minus 8

z2−8z−16
z squared minus 8 z minus 16

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, we have \(a = z\) and \(b = 4\). Applying the formula:

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

Now, let's compare this result with the provided options:

1. z² + 4z - 4z - 16: This simplifies to \(z^2 - 16\) (the \(4z\) and \(-4z\) cancel out).

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

3. z² - 8z - 16: This is \(z^2 - 8z - 16\), which is not equivalent.

4. z(z) + 4z(z) - 4(4): This simplifies to \(z^2 + 4z^2 - 16\), which is not equivalent.

The correct answer is the first option: z² + 4z - 4z - 16.