 The fundamental theorem of algebra is one of the quarterstones of mathematics. Unfortunately, it's not easy to prove using elementary methods. It can be done, it's just a lot more work. And so we usually prove it using complex analysis. So we have the following result. Every polynomial with real coefficients has a root of the form a plus b i. We can't actually prove this until we develop some complex function theory, but let's assume this is true for now and see where it takes us. So let f be a polynomial with real coefficients. Our theorem guarantees there is at least one complex root z equal to a plus b i. Consequently, f of z is equal to zero. But since f is a polynomial with real coefficients, the conjugate of f of z is f of z conjugate for all z. We can prove that directly. The conjugate of f of z is... Now, remember that f has real coefficients only, and so the conjugate of each of these coefficients is just the original coefficient. And so we get... which proves our result. But now let's consider. Since f of z is equal to zero, we can conjugate both sides, and the conjugate of zero is just zero, and the conjugate of f of z is f of z conjugate. And so f of z conjugate is zero as well, which proves an important theorem that complex roots of a real polynomial occur in conjugate pairs. Now, suppose f is a real polynomial. Let its real roots, if any, be r1 through rk. By the factor theorem, every real root corresponds to a linear factor x minus ri, and we can write f of x as a product of these linear factors and some other real polynomial. Now, if this other polynomial isn't a constant, then our theorem says that it will have at least one complex root. But since it's a real polynomial, the conjugate will also be a root. And since the number in its conjugate are roots, then their product is going to be a factor. Remember that the sum and product of a number in its conjugate are real, and so this product is also a real quadratic polynomial. And so that remaining polynomial has a quadratic factor. And remember it has complex roots, so it can't be factored into real factors. We call this an irreducible quadratic. And now Lather rins repeat to prove the fundamental theorem of algebra, at least one version. Every real polynomial can be written as a product of real, linear, and irreducible quadratic factors. Real factors correspond to real roots. The irreducible quadratic factors correspond by our assumption to complex roots of the form a plus b i. And this means that the roots of a real n-th degree polynomial are real or complex numbers. And this is important because... Well, let's take a look at that next.