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Reading some nice theorems about polynomials here induced me to wonder whether there are some more "standard" theorems about polynomials that I should know. Are there any similar and nice thereoms about polynomials? I am not working through any particular problem solving textbook, but rather solving online problems haphazardly. So I don't have any self-contained references handy.

My next question is specifically about one of these theorems in the given link. In each example, we used the given roots to construct a quadratic polynomial and from there used it to build the desired polynomial. For example, in the second problem we used the roots $3+i$ and $3-i$ to build the quadratic polynomial $x^2 - 6x + 10$ and from there built the desired cubic polynomial. I noticed that in each example the leading coefficient of the quadratic is $a=1$. Is this coincidence? Why isn't necessary to deduce that $a=1$? How do I know that there isn't some other set of roots such that the corresponding quadratic cannot have the leading term equal to $1$, so that I can't always blindly take $a=1$?

Now, I realize in these examples that, if one just blindly takes $a=1$, builds the quadratic from the two given roots, and then compute the roots, one will see that those roots coincide with the ones we began with. But how do I know this will always happen? I feel that there should be some justification/clarification of this point.

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    Vieta's formulas? Are you only working with quadratics? – 2017-02-13
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    @SimplyBeautifulArt I work with whatever polynomial comes my way! – 2017-02-13
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    Then it looks like you'll want to look up Vieta's formulas. – 2017-02-13
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    There is nobody "there" at the [here] -first line): broken link. – 2017-02-14
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    @JeanMarie Hmm...strange...I just clicked "here" (highlighted with blue) and it took me to the intended website. – 2017-02-14
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    Given some random polynomial usually we divide by the lead coefficient in order to have a lead coefficient of 1. This reduces complications especially considering the following. Given the roots $(x-c_1). (x-c_2), \ldots (x-c_n)$ when we take the product naturally the lead coefficient is 1. – 2017-02-14

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