Sum of Polynomial Roots (Proof)

Sum of Polynomial Roots

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