Question

which expression is equivalent to (z + 4) (z - 4)

a. z(z) + 4z(z) - 4(4)
b. z^2 + 4z - 4z - 16
c. 2z^2 - z^2 - 8
d. z^2 - 8z - 16

Answer 1

To find the expression equivalent to \( (z + 4)(z - 4) \), we can use the difference of squares formula, which states:

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

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

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

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

a. \( z(z) + 4z(z) - 4(4) \) - This evaluates to \( z^2 + 4z^2 - 16 = 5z^2 - 16 \), which is not equivalent.

b. \( z^2 + 4z - 4z - 16 \) - This simplifies to \( z^2 - 16 \), which is equivalent.

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

d. \( z^2 - 8z - 16 \) - This is not equivalent.

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

**b. \( z^2 - 16 \)**.