The Fundamental theorem of Algebra

the fundamental theorem of algebra fundamental I'll write it out theorem theorem of algebra tells us that if we have an nth degree polynomial so let's write it out so let's say I have let's say I have the function P of X and it's an it's defined by an nth degree polynomial so let's say it's a X to the n plus B X to the N minus 1 and you just go all the way to some constant term at the end so this is an nth degree polynomial the fundamental theorem of algebra tells us that this nth degree polynomial is going to have n exactly n roots n roots or another way to think about it they're going to be exactly n values for X which will make this polynomial make this expression on the right be equal to 0 so at first you might say okay that makes sense you've seen 2nd degree second degree polynomials whose graphs might look something like this so so let's see so y-axis that's the x-axis we know the second degree polynomial would define a parabola so it might look something like this and you could buy that okay this is a second degree that's 2nd degree and you see that this function equals a 0 exactly two places it has exactly it has exactly two roots it has two roots so that seems consistent with the fundamental theorem of algebra and you could also imagine a third degree polynomial looking like this so that's my y-axis this is my x-axis you could imagine a third degree polynomial looking something like this bamm-bamm I am and it just keeps going and here you see its third degree polynomial and you'll see it has one two three roots and I can have a fourth degree polynomial that maybe it looks something like this where it goes something like this and you say okay that makes sense it'll have one two three four roots but then you might start to remember things that don't always behave in this way for example many many many many times we've seen parabolas we've seen second degree polynomials that look more like this where they don't seem to intersect the x-axis so this seems this seems to conflict with the fundamental theorem of algebra the fundamental theorem of algebra says if we have a second degree second degree polynomial then we should have exactly two roots now this is the key the fundamental theorem of algebra it extends our number system we're not just talking about real roots we're talking about complex roots and in particular the fundamental theorem of algebra allows even these coefficients to be complex and so when we're looking at these these first examples these were all real roots and real numbers are a subset of complex numbers so here you had two real roots here you had three real roots in this orange function you had four real roots in this yellow function this yellow parabola right over here the second-degree polynomial we have no real roots that's why you don't see it intersecting the x-axis but we will have two complex roots so this one right over here will have two complex two complex roots and the complex roots the the non-real complex roots because really real numbers are a subset of complex numbers these always come in pairs and we'll see that in future videos so for example if you have if you have a third degree polynomial it might look something like this a third degree problem it might look something like this where it has one real root but then the fundamental theorem of algebra tells us it necessarily has two other roots because it is a third third degree so we know that the other two roots must be non-real complex roots now could you have a situation where you have a third degree polynomial with three complex roots so can you have three non non real complex non-real complex roots is this possible for a third degree polynomial well the answer is no because complex roots as we'll see in the next few videos always come in pairs that they're coming in pairs where they are conjugates of each other so you could have you could have a fourth degree you could have a fourth degree polynomial that has no real roots for example something might look something like this in this case you would have two pairs of complex roots or you would have four non-real complex roots and you could group them into two pairs where in each pair you have conjugates and we'll see that in the next video