Solve the equation by factoring and using the zero product principle.

16 y^3 + 6 = y + 96 y^2

Question

Solve the equation by factoring and using the zero product principle. 

16 y^3 + 6 = y + 96 y^2

anonymous · Answer

Did you try synthetic division with possible rational zeros of the form of factors of 6 over factors of 16?

anonymous · Answer

Here's what I did.

anonymous · Answer

16y^3-96y^2=y-6

anonymous · Answer

Then I factored the LHS: 16y^2(y-6) = (y-6)

anonymous · Answer

Then I basically just guessed that 6 was one of the answers, and it turned out I was right, but other than that I'm pretty lost.

anonymous · Answer

You have to get all of the terms over on one side with the other side equalling 0.  That's the starting point.

anonymous · Answer

I thought so, I just...have no idea how to factor 16y^3-96y^2-y+6

anonymous · Answer

It's really a brute-force method.  You have to write out the candidate possibilities for the zeros.  As I mentioned above, the candidates will be o f the form of factors of 6 over factors of 16.  So the factors of 6 wil + and - of 1,2,3,6.  Similar for the denominator but taking 16.  Lots of combinations, but they do whittle down quickly in practice.

anonymous · Answer

Do you have any idea how to solve it using factoring, even though that method might come more naturally to you?

anonymous · Answer

So possibilities are + and - of 1/16, 1/8, 1/4, 1/2, 1, 2, 3/16, etc.  In practice, when these problems are given as homework, they usually are the simpler numbers, so try the integers first.  They don't want to make it too hard.  I actually solved the problem as I'm describing it.  Big hint: the answer is one of the integers.

anonymous · Answer

I know 6 is one of the answers. None of the other integers are working.

anonymous · Answer

So, one of the zeros is an integer.  The other 2 zeros are real numbers but one is a positive irrational # and the other is a negative irrational #.

anonymous · Answer

Ok, you did really well to find that 6!  Good job!

anonymous · Answer

You are then left with a quadratic equation. And when you have that, you don't have to synthetic division anymore.  You now use the general solution to the quadratic.

anonymous · Answer

Alright, thank you very much.

anonymous · Answer

you're welcome and good luck to you on all all your studies.