# Fundamental theorem of algebra | Polynomial and rational functions | Algebra II | Khan Academy

Basic theorem of algebra. The main – I will write this – theorem of algebra tells us that if we have a polynomial of degree n … let’s write this down. Let’s say I have the function p (x) and it is expressed by a polynomial of degree n. Let’s say it’s ah of degree n plus bx of degree (n – 1) and you just move on to some constant at the end. This is a polynomial of degree n. The basic theorem of algebra tells us that this polynomial of degree n will have exactly n root or another way to think about it is that there will be exactly n values \u200b\u200bfor x, which will make this polynomial, this expression on the right should be equal to 0. The first thing you might think is that this makes sense. We’ve seen second-degree polynomials, whose graphs may look similar to this. This is the y-axis, and this is the x-axis. We know that the graph of a second degree polynomial will be a parabola, so it may look like this and you will agree that it is correct. This is the second degree, this is the second degree, you see that this feature is equal to 0 in exactly two places. It has exactly two roots. There are two roots, so this seems compatible with the basic theorem of algebra. And you can also imagine that the graph of a third degree polynomial will look like this … This is the y-axis. This is the x-axis. You can imagine that a polynomial of the third degree it will look like this. Bam, bam, bam and on. Here you see that it is a polynomial of the third degree, because there are 1, 2, 3 roots. I can also draw the graph of a polynomial of the fourth degree. Maybe it will look like this, and you will tell yourself that this is logical. There will be 1, 2, 3, 4 roots. But then you can start remembering things, who do not always behave in this way. For example, many, many, many times we have seen parables, we have seen second degree polynomials, which look more like what do not appear to intersect the x-axis. This seems to contradict the basic theorem of algebra. The basic theorem of algebra says that if we have a polynomial of second degree, then we must have exactly two roots. This is the key thing here. The basic theorem of algebra expands our number system. We’re not just talking about real roots, we are talking about complex roots, and, above all, the basic theorem of algebra allows even these coefficients to be complex. When we look at these first examples, all of these are real roots and real numbers are a subset of complex numbers. There were 2 real roots here. There were 3 real roots here. There were 4 real roots in this orange graphic. On this yellow graph (of function), of this yellow parabola here – the graph of a polynomial of the second degree – we have no real roots. That’s why you don’t see intersections with the x-axis, but we will have 2 complex roots. This here will have 2 complex roots. Complex roots – those of them that are not real, because real numbers are a subset of complex numbers – they always come in pairs and we will see this in future videos. For example, if you have a third degree polynomial, it may look like this. The graph of a third degree polynomial may look like this, where there will be 1 real root, but then the basic theorem of algebra tells us that there needs to be 2 more roots, because he’s third degree, so we know that the other 2 roots must not be from the real complex roots. Could there be a situation in which to have third degree polynomial with 3 complex roots? Can you have 3 complex roots that are not real? Is this possible for a third degree polynomial? The answer is no, because the complex roots, as we will see in the next few videos, they always come in pairs. They come in pairs and are conjugated. You can have a polynomial of the fourth degree, which has no real roots. The graphics may look like this. In this case you will have two pairs of complex roots or you will have 4 complex roots that will not be real. And you can group them into two pairs, in which each pair will be conjugated and we’ll see that in the next video.