What I want to do in this video is find out if there is any fast way
to figure out the sum of the roots of any polynomial
And there actually is and that's why I'm doing this video
So lets start with a second degree polynomial
so lets say it's x squared plus a one x plus a two is equal to zero
so this is just a standard quadratic equation right here, second degree equation
and you might be saying hey but you put the coefficent
on the x term equal to one
and in general you can always convert any polynomial
and I'll do it with a second degree but if you have
A x squared plus B x plus C is equal to 0
then you can just divide both sides of this equation by A
and you're going to have something that's in this form
where the coefficent on the x squared term
is going to be equal to one.
And you can do that with any polynomial that's set equal to 0
so with that out of the way let's think about what the
sum of the roots of this are going to be
so this is a second degree polynomial
it's a quadratic equation, so it'll have two roots
they can be real or complex
so let's call the roots R one and R two
and that tells us that these are roots that
X minus R 1 times X minus R 2
is going to be equal to 0
and if we multiply this out we get
X times X is X squared
X times negative R2 is negative R2X
and then we have negative R1 times X and negative R1X
and then we have negative R1 times negative R2
which is plus R1R2 is equal to 0
and we can simplify this middle term a little bit
it becomes x squared minus R1 plus R2