the binomial 8y^12+z^6 can be factored into a binomial and a trinomial. the binomial factor is (2y^4+z^2). whats the trinomial?

Just do the long division to get

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

or, you could recognize it as the sum of two cubes:

8y^12+z^6
= (2y^4)^3 + (z^2)^3
= (2y^4 + z^2)((2y^4)^2 - (2y^4)(z^2) + (z^2)^2)
= (2y^4 + z^2)(4y^8 - 2y^4z^2 + z^4)

To determine the trinomial factor, we can use polynomial division. Here's how you can find it by dividing the given binomial by the binomial factor we already have.

Step 1: Set up the division:
Divide (8y^12 + z^6) by (2y^4 + z^2).

4y^8 + 2y^4z^4
-------------------------------------
2y^4 + z^2 | 8y^12 + 0y^10 + 0y^8 + 0y^6 + 0y^4 + z^6

Step 2: Divide the highest degree term:
Divide 8y^12 by 2y^4, which gives 4y^8.

4y^8 + 2y^4z^4
-------------------------------------
2y^4 + z^2 | 8y^12 + 0y^10 + 0y^8 + 0y^6 + 0y^4 + z^6

Step 3: Multiply the binomial factor by the result from the previous step:
Multiply (2y^4 + z^2) by 4y^8 to get 8y^12 + 4y^8z^2.

4y^8 + 2y^4z^4
-------------------------------------
2y^4 + z^2 | 8y^12 + 0y^10 + 0y^8 + 0y^6 + 0y^4 + z^6
- (8y^12 + 4y^8z^2)
-------------------------------------
- 4y^8z^2 + z^6

Step 4: Bring down the next term:
Bring down -4y^8z^2.

4y^8 + 2y^4z^4
-------------------------------------
2y^4 + z^2 | 8y^12 + 0y^10 + 0y^8 + 0y^6 + 0y^4 + z^6
- (8y^12 + 4y^8z^2)
-------------------------------------
- 4y^8z^2 + z^6
- (-4y^8z^2)
-------------------------------------
z^6

Step 5: Divide the new term:
Divide z^6 by 2y^4, which gives (1/2)y^2z^2.

4y^8 + 2y^4z^4 + (1/2)y^2z^2
------------------------------------------------
2y^4 + z^2 | 8y^12 + 0y^10 + 0y^8 + 0y^6 + 0y^4 + z^6
- (8y^12 + 4y^8z^2)
-------------------------------------
- 4y^8z^2 + z^6
- (-4y^8z^2)
-------------------------------------
z^6 + (1/2)y^2z^2

At this point, we cannot divide further, so we stop.

Therefore, the trinomial factor is (4y^8 + 2y^4z^4 + (1/2)y^2z^2).

Hence, the factored form of the binomial 8y^12 + z^6 into a binomial and a trinomial is:
(2y^4 + z^2)(4y^8 + 2y^4z^4 + (1/2)y^2z^2).