Question

which is equivalent to ((4xy - 3z)^2), and what type of special product is it? (\bigcirc) (16x^2y^2 + 9z^2), the difference of squares (\bigcirc) (16x^2y^2 + 9z^2), a perfect square trinomial (\bigcirc) (16x^2y^2 - 24xyz + 9z^2), the difference of squares (\bigcirc) (16x^2y^2 - 24xyz + 9z^2), a perfect square trinomial

Explanation:

Step1: Expand \((4xy - 3z)^2\)

Using the formula \((a - b)^2 = a^2 - 2ab + b^2\), where \(a = 4xy\) and \(b = 3z\).
\[

$$\begin{align*} (4xy - 3z)^2&=(4xy)^2 - 2\times(4xy)\times(3z)+(3z)^2\\ &=16x^{2}y^{2}-24xyz + 9z^{2} \end{align*}$$

\]

Step2: Identify the type of polynomial

A perfect square trinomial has the form \(a^2 - 2ab + b^2\) (or \(a^2+2ab + b^2\)). Here, \(16x^{2}y^{2}-24xyz + 9z^{2}\) matches \(a^2 - 2ab + b^2\) with \(a = 4xy\) and \(b = 3z\), so it is a perfect square trinomial.

Answer:

\(16x^{2}y^{2}-24xyz + 9z^{2}\), a perfect square trinomial (the fourth option)