Final answer:
To make the trinomial 16y + * + 81 a perfect square, we need to find a monomial that can be inserted in the middle term to make it a perfect square trinomial. We can do this by identifying the perfect square trinomial's form and determining the values for the first and last terms. Then, we can find the middle term using the formula for a perfect square trinomial.
Step-by-step explanation:
To make the trinomial 16y + * + 81 a perfect square, we need to find a monomial that can be inserted in the middle term to make it a perfect square trinomial.
Let's start by identifying the perfect square trinomial. It has the form (a + b)² = a² + 2ab + b². In this case, the trinomial is 16y + * + 81. To match the form (a + b)², the first term must be a square of some monomial, which is (4y)² = 16y², and the last term must also be a square, which is (9)² = 81.
Now, we can find the middle term by using the formula 2ab, where a and b are the terms in the perfect square trinomial. In this case, a is 4y and b is 9. Therefore, the middle term is 2(4y)(9) = 72y.
So, to make the trinomial 16y + * + 81 a perfect square, we can insert the monomial 72y in the middle, resulting in the trinomial (4y + 72y + 9)².