Number Sense Questions

[mathperson]

Could someone please tell me the formulas for finding the product/sum/sum of roots taken two at a time/etc for a quadratic and cubic equation?

[BlasianAsian]

Look at the thread titled, "Mas Preguntas". Lol, we just answered that!

[mathperson]

Thank You!

[PokémonMaster]

In the post "Tengo Mas Preguntas",it was only explained that the sum of the product of the roots taken two at a time was c/a, so I will explain the other two.
To take the sum of the roots, take b/a. For the product of the roots, take the constant over a.
Example: x^3-6x^2+11x-6. the sum of the roots is -6/1.The product of the roots is also -6/1. The three factors of this equation, (x+1),(x+2), and (x+3) can be used to prove it. I hope I explained it clearly enough.

[nsguy1350]

I made a huge post about this, so I guess I'll make it again. I'll put it up shortly.
For some reason, I posted it but it didn't go through, and I didn't notice until today.

[nsguy1350]

Alright, I want you to see the patterns associated with sums/product/sum of the products taken X at a time. First, I want you to see the order that they come in, along with the alternating signs rule.
Before that though, just a quick rule you should remember:
sum = -b/a, product = [unparseable or potentially dangerous latex formula]. Therefore, all products for even degree polynomials are just the constant/a, while odd degrees use the negative constant/a rule.

Let me see if you can see the pattern in this through the chart. Abbreviations: sum = sum of roots, sopX = sum of the product of the roots taken X at a time, and prod = product of the roots.

linear: [unparseable or potentially dangerous latex formula]
sum = -b/a
quadratic: [unparseable or potentially dangerous latex formula]
sum = -b/a prod = c/a
cubic: [unparseable or potentially dangerous latex formula]
sum = -b/a sop2 = c/a prod = -d/a
quartic: [unparseable or potentially dangerous latex formula]
sum = -b/a sop2 = c/a sop3 = -d/a prod = e/a
And so on...
You can see that you always get the pattern: -b/a,c/a,-d/a,e/a,... until you run out. That's what I call the alternating signs rule.
Also, notice that sopXs go up as you go right: sop2 is c/a, while sop3 is -d/a, and so on.

Now, to see a really nice pattern, notice how sum comes before sop2 and prod comes after sopX, the largest sop.
The sum of thee roots is really the sum of the product of the roots taken one at a time, and the product is just the sum of the product of the roots taken the maximum amount of times.
Then, you can see the pattern: sop1 (sum) --> sop2 --> sop3 --> sop4 --> ... ---> sopX (prod). For a lame name for this, how about the increasing amount rule? I don't know the names for these rules or if anyone has found them; the alternating signs is pretty easy to see, and so is the increasing amount rule if you notice the pattern, but I just found out the sum/prod special cases yesterday, while I was making the post that was responding to this that didn't go through. If you keep these in mind, it should stick better for you.
Hope that helps!
Also, PokemonMaster, I believe they would both be positive 6/1 = 6, because sum is always negative, while product for a cubic is negative as well.
[unparseable or potentially dangerous latex formula] is very different from [unparseable or potentially dangerous latex formula] keep that in mind!

[PokémonMaster]

Hmm... I guess you're right.I was looking at it the wrong way, I guess. Thank you for the correction.

[mathperson]

Thank you nsguy! This makes a lot of sense.