 Descartes' method of tangents relied on an equation having a repeated root. Fermat's method of tangents and optimization also relied implicitly on these repeated roots. But how can we identify the conditions required to have a repeated root without having to do a lot of painful algebra? A solution came from two letters of July 15, 1657 and February 6, 1658, written by Jan Heude. Heude's procedure relied on finding the greatest common divisor of two polynomials. To do so, Heude used the following procedure. Set both polynomials equal to zero, solve the lower degree equation for the highest power, substitute into the other, and lather, rinse, repeat. The last polynomial equation corresponds to the greatest common divisor. For example, let's find the GCD of these two polynomials. So first, we'll set both polynomials equal to zero and solve the lower degree equation for the highest degree term. So we'll solve the quadratic for the x squared term. Now, we can substitute for x squared 11x minus 24. But what about the x cubed term? To begin with, we might note that if we multiply all of our terms by x, we get. And while this expression has an x squared in it, we know what x squared is equal to. So we can substitute and simplify. And that means our x cubed term can be replaced by a linear term. So we replace x squared with 11x minus 24 and x cubed with 97x minus 264. The other terms remain and simplify. Now, since we're looking for the GCD of two polynomials, we can actually factor out any common constant term. So let's do that. It's helpful to think about this new polynomial as replacing our original polynomial. And at this point, we're back where we started, except now we have two different polynomials. Well, now we have two polynomial equations, the quadratic and the linear. So we'll set both polynomials equal to zero and solve the lower degree equation for the highest degree term. And so that's this linear equation x minus eight. We'll solve it getting x equal to eight. And if we substitute x equal to eight into our other polynomial, we get nothing, which means that x minus eight is a factor of x squared minus 11x plus 24. And so it's going to be the GCD. Now we won't go into the details, but this is essentially a version of the Euclidean algorithm. The ability to find the GCD of two polynomials was important for HUD's main result. First, we present HUD's first rule. Let x be a polynomial with a repeated root. If the terms of x are multiplied by any arithmetic regression, the new polynomial y will have a root in common with the original. HUD includes a proof in his second letter. For example, we might use the rule as follows. Let's try to factor this polynomial. And so we note that if our arithmetic expression includes a zero, we'll be able to eliminate a term. Now we choose completely at random the arithmetic progression one, zero, negative one, negative two, negative three, and multiplying our terms. And HUD's theorem says that if this original polynomial has a repeated root, then that repeated root will be a common factor with our new polynomial. So we'll find the GCD of our two polynomials, which turns out to be x minus three. That means x minus three is a factor of our original polynomial, and in fact it's a repeated root. Our original polynomial is x minus three squared times another polynomial, which turns out to be unfactorable. And so here's our factorization. In his second letter, HUD notes that the method can be used to find the maximum and minimum values of polynomial expressions. So to find the maximum or minimum of a polynomial, set it equal to z and multiply its terms by an arithmetic sequence. The new polynomial will have roots in common with the original, and one of the roots will correspond to a maximum or minimum. Let's see why this works. If we imagine graphing our polynomial, y equals z will intersect the curve several times at non-extrama. But if I imagine changing the value of z, that moves this horizontal line vertically, and at a maximum or minimum value, y equals z will be tangent to the curve, and the point of tangency corresponds to a repeated root. So let's try to find the extreme values of this expression. Suppose the extreme value is x cubed minus 6x squared minus 15x plus 250 equals z. We can rearrange, and we want this equation to have a repeated root. Now while we could pick any arithmetic sequence we want, since we don't know the value of z, we might as well make that last term 0. That way our HUD polynomial will not incorporate this unknown value. So we take any arithmetic sequence that ends in 0, how about the simplest 3, 2, 1, 0. Multiply the terms of our expression by the terms in the arithmetic sequence to get the HUD polynomial, and by HUD's theorem, the HUD polynomial should have a root in common with the original. So while we could find the greatest common divisor of the two polynomials, in practice it'll probably be just easier to find the roots of our HUD polynomial. So we'll factor, and so our possible repeated roots are x equals 0, 5, and negative 1. And we'll check them out. If x equals 0, then z is equal to 250, but our equation does not have a repeated root. So we see that x equals 0 does not give us an extreme value. If x equals 5, then z is equal to 150, and solving our equation. Now remember we know that x equals 5 is a solution, so x minus 5 is a factor. So our left hand side is x minus 5 times something, which works out to be, and in fact our something factors in 2. So we see that x equals 5 is a repeated root, and so x equals 5 will correspond to an extreme value. Now by other means, we'll find out that this corresponds to a local minimum value. If x equals negative 1, z works out to be, again solving our equation. If x equals negative 1 is a root, then x plus 1 must be a factor, and so our left hand side is x plus 1 times, and then, so we see that x plus 1 is a repeated factor, and x minus 1 is a repeated root, so there will be an extreme value at x equal to negative 1, and again through other means, we can see that this works out to be a local maximum.