Section 1.1. Classical Formulas 3 for we have just seen, in Example 1.2, that an “honest” real and positive root can appear in terms of such radicals: 3 1 2 −6 + −400 27 + 3 1 2 −6 − −400 27 is an integer! Thus, the cubic formula was revolutionary. For the next 100 years, mathematicians reconsidered the meaning of number, for understanding the cubic formula raises the questions whether negative numbers and complex numbers are legitimate entities. Consider the quartic F (X) = X4 +bX3 +cX2 +dX +e. The change of variable X = x − 1 4 b yields a simpler polynomial f(x) = x4 + qx2 + rx + s whose roots give the roots of F (x): if u is a root of f(x), then u − 1 4 b is a root of F (x). The quartic formula was found by Luigi Ferrari in the 1540s, but we present the version given by Descartes in 1637. Factor f(x), f(x) = x4 + qx2 + rx + s = (x2 + jx + )(x2 − jx + m), and determine j, and m [note that the coeﬃcients of the linear terms in the quadratic factors are j and −j because f(x) has no cubic term]. Expanding and equating like coeﬃcients gives the equations + m − j2 = q, j(m − ) = r, m = s. The first two equations give 2m = j2 + q + r/j, 2 = j2 + q − r/j. Substituting these values for m and into the third equation yields the resolvent cubic: (j2)3 + 2q(j2)2 + (q2 − 4s)j2 − r2. The cubic formula gives j2, from which we can determine m and , and hence the roots of the quartic. The quartic formula has the same disadvantage as the cubic formula: even though it gives a correct answer, the values of the roots are usually unrecognizable. Note that the quadratic formula can be derived in a way similar to the deriva- tion of the cubic and quartic formulas. The change of variable X = x − 1 2 b re- places the quadratic polynomial F (X) = X2 + bX + c with the simpler polynomial f(x) = x2 + q whose roots give the roots of F (x): if u is a root of f(x), then u − 1 2 b is a root of F (x). An explicit formula for q is c − 1 4 b2, so that the roots of f(x) are, obviously, u = ± 1 2 √ b2 − 4c thus, the roots of F (X) are 1 2 ( − b ± √ b2 − 4c ) . It is now very tempting, as it was for our ancestors, to seek the roots of a quintic F (X) = X5 + bX4 + cX3 + dX2 + eX + f (of course, they hoped to find roots of polynomials of any degree). Begin by changing variable X = x− 1 5 b to eliminate the X4 term. It was natural to expect that some further ingenious substitution together with the formulas for roots of polynomials of lower degree, analogous to the resolvent cubic, would yield the roots of F (X). For almost 300 years, no such formula was found. In 1770, Lagrange showed that reasonable substitutions lead to a polynomial

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