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

Question

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

The expression \((z + 4)(z - 4)\) is a difference of squares, which can be simplified using the formula \(a^2 - b^2 = (a + b)(a - b)\).

Here, \(a = z\) and \(b = 4\). Therefore:

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

Now, let's evaluate the responses given:

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

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

3. z² - 8z - 16: This is not equivalent to the original expression.

4. z(z) + 4z(z) - 4(4): This simplifies to \(z^2 + 4z^2 - 16 = 5z^2 - 16\), which is not equivalent to the original expression.

The correct response that is equivalent to \((z + 4)(z - 4)\) is:

z² + 4z - 4z - 16 (simplifies to \(z^2 - 16\)).