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Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,What I want to do in this video is find out if there is any fast way
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,to figure out the sum of the roots of any polynomial
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,And there actually is and that's why I'm doing this video
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,So lets start with a second degree polynomial
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,so lets say it's x squared plus a one x plus a two is equal to zero
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,so this is just a standard quadratic equation right here, second degree equation
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and you might be saying hey but you put the coefficent
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,on the x term equal to one
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and in general you can always convert any polynomial
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and I'll do it with a second degree but if you have
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,A x squared plus B x plus C is equal to 0
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,then you can just divide both sides of this equation by A
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and you're going to have something that's in this form
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,where the coefficent on the x squared term
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,is going to be equal to one.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,And you can do that with any polynomial that's set equal to 0
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,so with that out of the way let's think about what the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,sum of the roots of this are going to be
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,so this is a second degree polynomial
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,it's a quadratic equation, so it'll have two roots
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,they can be real or complex
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,so let's call the roots R one and R two
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and that tells us that these are roots that
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,X minus R 1 times X minus R 2
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,is going to be equal to 0
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and if we multiply this out we get
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,X times X is X squared
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,X times negative R2 is negative R2X
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and then we have negative R1 times X and negative R1X
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and then we have negative R1 times negative R2
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,which is plus R1R2 is equal to 0
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and we can simplify this middle term a little bit
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,it becomes x squared minus R1 plus R2