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

Question

z^2 + 4z - 4z - 16
2z^2 - z^2 - 8
z^2 - 8z - 16
z(z) + 4z(z) - 4(4)

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

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

Now let's evaluate the options provided:

1. \(z^2 + 4z - 4z - 16\):

   • Simplifying this gives \(z^2 - 16\), which matches our result.

2. \(2z^2 - z^2 - 8\):

   • Simplifying this gives \(z^2 - 8\), which does not match.

3. \(z^2 - 8z - 16\):

   • This does not match either, as it has an incorrect linear term.

4. \(z(z) + 4z(z) - 4(4)\):

   • Simplifying this gives \(z^2 + 4z^2 - 16 = 5z^2 - 16\), which also does not match.

Therefore, the correct expression that is equivalent to \((z + 4)(z - 4)\) is:

\[ z^2 - 16 \]

Out of the options provided, the first option \(z^2 + 4z - 4z - 16\) simplifies correctly to \(z^2 - 16\) and is equivalent to the expression we derived.