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

z2−8z−16
z squared minus 8 z minus 16

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

2z2−z2−8

Answer 1

To find an 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, we can identify \(a = z\) and \(b = 4\). Therefore, applying the formula, we have:

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

Now, let's examine the options given:

1. \(z^2 + 4z - 4z - 16\) simplifies to \(z^2 - 16\).
2. \(z^2 - 8z - 16\) is not equivalent.
3. \(z(z) + 4z(z) - 4(4)\) is not simplified to the right form.
4. \(2z^2 - z^2 - 8\) simplifies to \(z^2 - 8\), which is also not equivalent.

Thus, the expression that is equivalent to \((z+4)(z−4)\) is:

\(z^2 - 16\)

None of the options list \(z^2 - 16\) directly, but the first option is the one that simplifies to it after combining like terms.

So, the best response from the provided options is:

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