Question

Which expression is equivalent to (z+4)(z–4)?

A. 2z^2–z^2–8
B. z^2+4z–4z–16
C. z(z)+4z(z)–4(4)
D. z^2–8z–16

Answer 1

To find an equivalent expression 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, we can identify \(a = z\) and \(b = 4\). Thus, we can apply the formula:

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

Now we can analyze the options provided:

A. \(2z^2 - z^2 - 8\) simplifies to \(z^2 - 8\) (not equivalent).
B. \(z^2 + 4z - 4z - 16\) simplifies to \(z^2 - 16\) (equivalent).
C. \(z(z) + 4z(z) - 4(4) = z^2 + 4z^2 - 16 = 5z^2 - 16\) (not equivalent).
D. \(z^2 - 8z - 16\) (not equivalent).

The only option that simplifies to \(z^2 - 16\) is:

**B. \(z^2 + 4z - 4z - 16\)**.