Question

65.b insert a monomial into each trinomial such that the result is a perfect square. 16y² +.... + 81 answer 16y² + + 81

Explanation:

Step1: Recall perfect square trinomial formula

A perfect square trinomial is of the form \((a + b)^2 = a^2 + 2ab + b^2\) or \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a^2 = 16y^2\), so \(a = 4y\) (since \((4y)^2 = 16y^2\)), and \(b^2 = 81\), so \(b = 9\) (since \(9^2 = 81\)).

Step2: Calculate the middle term

Using the formula for the middle term of a perfect square trinomial (\(2ab\) or \(-2ab\)), we substitute \(a = 4y\) and \(b = 9\). So \(2ab = 2\times4y\times9 = 72y\) or \(-2ab=-72y\). Since the problem asks for a monomial, either \(72y\) or \(-72y\) works, but typically we consider the positive middle term for such problems unless specified otherwise. So the middle term should be \(72y\) (or \(-72y\), but \(72y\) is more common here).

Answer:

\(72y\) (or \(-72y\), but \(72y\) is the typical answer for this form)