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The Fundamental theorem of Algebra

Sal introduces the Fundamental Theorem of Algebra, which, as is reflected in its name, is a very important theorem about polynomials. Created by Sal Khan.

Video transcript

Voiceover:The fundamental theorem of algebra. Fundamental, I'll write it out, 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 the function p of x and 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 one 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, or another way to think about it, there are going to be exactly n values for x, which will make this polynomial, make this expression on the right, be equal to zero. So at first you might say, "Okay, that makes sense." You've seen second-degree polynomials, whose graphs might look something like this. So, that's the y axis and that's the x axis. We know a 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 second degree, and you see that this function equals zero at exactly two places. 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. Bam, bam, bam, and it just keeps going. And here you see it's a third-degree polynomial, and you'll see it has one, two, three roots. And I could have a fourth-degree polynomial. Maybe it looks something like this, where it goes something like this, and you say, "Okay, that makes sense." It will 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 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 to conflict with the fundamental theorem of algebra. The fundamental theorem of algebra says if we have a 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 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 roots. And the complex roots, 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 a third-degree polynomial, It might look something like this. A third-degree polynomial might look something like this, where it has one real root, but then the fundamental theorem of algebra tell us it necessarily has two other roots because it is a 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-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. They're coming in pairs where they are conjugates of each other. So, 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.