Expert: Scott A Wilson - 2/15/2010

Question
I have one problem I can't figure out.  It is a "solving equations by factoring" question.

The problem is: -t^3-3t^2+4t+12.  The answer is -3,-2,2.  But I can't figure out how to get there.  

I mostly just need the set-up as this is the only problem on my homework with 4 terms and I don't see any GCFs or grouping ways to factor it.

Answer
When you are given -t³ - 3t² + 4t + 12, this can be factored in pieces.
Note that it is -t²(t+3) + 4(t+3), so it can be rewritten as (-t²+4)(t+3).

Next, the first term is the difference of two squares,
and that factors into -(t-2)(t+2), so the whole problem becomes -(t-2)(t+2)(t+3).

It can then be seen to have roots 2, -2, and -3.


It can also be rearranged into  -t³ + 4t - 3t² + 12, which is -t(t²-4) - 3(t²-4),
and that reduces to -(t+3)(t²-4).  Again, we just factored t²-4.