Hey xXCandyDXx! Glad Jake could be of help!
In addition, a tip for spotting when to use the "sums of roots" formulae. Pretty much, if a question follows the following template:
Find the roots of ____________________ given that __________________ or even
The roots of ___________________ form an arithmetic series. Find the value of k (some unknown in the polynomial)) Essentially, if you are given a polynomial with something to find, and all you are given is some information about the roots, chances are that these formulae will help you. They are very powerful in surprising ways.
While I'm here, I have an issue with the method you posted above. It's very possible I am misunderstanding, but it seems like the notions of
root and
factor are being confused. Remember, factors can be multiplied together to get back to the original, roots are just solutions to the polynomial equal to zero.
So, in the first few lines, we multiply the roots together and then form an identity with the original polynomial. This doesn't make sense to me. We could do it with factors, but multiplying the roots of a polynomial together has no relationship with the original, besides the relationships highlighted in the formulae used by Jake's method.
The factor rule makes perfect sense. The first part of the solution however, while the correct answer is obtained, does not make as much sense to me, and I would argue that it is incorrect. I'd love to hear others interpretations on the matter!