Question

which is equivalent to (4xy - 3z)², and what type of special product is it?
○ 16x²y² + 9z², the difference of squares
○ 16x²y² + 9z², a perfect square trinomial
○ 16x²y² - 24xyz + 9z², the difference of squares
○ 16x²y² - 24xyz + 9z², a perfect square trinomial

Explanation:

Step1: Recall the formula for squaring a binomial

The formula for \((a - b)^2\) is \(a^2 - 2ab + b^2\). Here, \(a = 4xy\) and \(b = 3z\).

Step2: Calculate \(a^2\)

\(a^2=(4xy)^2 = 16x^{2}y^{2}\)

Step3: Calculate \(2ab\)

\(2ab = 2\times(4xy)\times(3z)=24xyz\)

Step4: Calculate \(b^2\)

\(b^2=(3z)^2 = 9z^{2}\)

Step5: Substitute into the formula

\((4xy - 3z)^2=a^2-2ab + b^2=16x^{2}y^{2}-24xyz + 9z^{2}\). A perfect square trinomial is of the form \(a^2-2ab + b^2\) (or \(a^2 + 2ab + b^2\)) which is the square of a binomial, so this is a perfect square trinomial.

Answer:

D. \(16x^{2}y^{2}-24xyz + 9z^{2}\), a perfect square trinomial (assuming the last option is D, if the options are labeled as A, B, C, D with the last one being this, the answer is the option with \(16x^{2}y^{2}-24xyz + 9z^{2}\) and "a perfect square trinomial")