 Around 1745, Euler mentioned he had a proof of the fundamental theorem of algebra, but failed to publish it. In response, Jean-Léron d'Alembert published his own proof in 1746. D'Alembert's proof relied on several questionable assumptions, so it is not typically regarded as an actual proof. However, it used some ideas that were foreshadowed the modern proof using complex analysis. Euler eventually published his own proof in 1749 as researchers on the imaginary roots of equations. Euler's proof relies on some elementary results from the theory of equations. The most important one is that given the factorization of a polynomial, the coefficients of the polynomial can be determined by sums of products of the roots. A few others, an odd-degree polynomial has at least one real root, and an even-degree polynomial with a negative constant term has at least two real roots, one positive and one negative. Euler also made use of the fact that in an nth-degree equation, the n-first-degree term can be eliminated using the substitution y equals 1nth-a. So let's assume we've already done that to an arbitrary quartic, which gives us a depressed quartic, one without a cubic term, and suppose we wanted to express this as the product of two quadratic factors. The factorization must be of the form where our x coefficients are equal and opposite, and comparing the product with the given equation gives us, and since we have three equations and three unknowns, we can find alpha, beta, and u. Now it turns out to be convenient if we find alpha and beta first, because their product is equal to d, which is the constant in our original equation. And so we find, so we know two alpha and two beta, so if we multiply them together we get, and remember the product alpha beta is equal to d, the constant in the original equation. So we can simplify. Note that the unknown is u, while the coefficients are expressions of the coefficients of the original equation. We'll eventually call this a Resolvent equation. What's important in this case is this is an even-degree polynomial with a negative constant, so it must have a real root. And consequently, Euler conclude, real quartics can always be factored into a product of two real quadratics. Now this suggests the next step might be to prove that an eighth-degree polynomial with real coefficients can be factored into a product of fourth-degree polynomials with real coefficients. And Euler does outline those steps. So again we'll begin with a depressed octic, while lacking the seventh-degree term, and we'll assume factors of this form where because their product doesn't have a seventh-degree term, their cubic terms have to be equal and opposite. And we can compare coefficients and obtain a system of seven equations in seven unknowns. Unfortunately, these are non-linear equations, and Euler recognized that it would be very difficult and perhaps impossible to solve the equation by which the unknown u is found. And so we need another approach. And Euler gives it, and we'll talk about that next.