An introduction to the modern theory of equations, by Florian Cajori.

ELIMIN ATION 97 The discriminants of the general quadratic, cubic, and quartic are, respectively, as follows: Quadratic disc. (bl2 - bob,); bo Cubic disc., ~ 35 - 27 (G + 4 H); b06 Quartic disc., ~ 51, = 6(13 - 27 J2). 77. Discriminant expressed as a Symmetric Function of the Roots. Since the discriminant of the equation f(x) == 0 vanishes always when at least two roots are equal, but under no other conditions, it follows that a, - c2 must be a factor of the discriminant. For if a, and ac are the equal roots, a, - (a2 is the only simple factor which will vanish because of this equality. But an interchange of any two roots, say a, and tl must not alter the numerical value or the sign of the discriminant, since the discriminant is a constant when the coefficients of the equation are constants. Hence the lowest positive power to which the factor aC - ac can occur in the discriminant is the second power. In other words, (aI - a,)2 is a factor of the discriminant. Since this reasoning applies to any two roots whatever, (a1 - aQ)2 is a factor; also (a1 - C4)2; and so on. Hence the product D n (, - a)2 ( (a1 - a )... (a- )2 is a factor of the discriminant. If the multiplications indicated in this product were carried out, each term would be of the n(n - 1)th degree in the roots. The resultant of f(x) = 0 and f'(x) = 0 may be expressed by ~ 73 as f'(cl). f'(a) **f'(9f ) where (,, a2,., an are the roots of.f(x) =. One term of this product is (na0o2)m(ja]2 * 9,)"-l1; the degree of tis term in the roots H

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Title
An introduction to the modern theory of equations, by Florian Cajori.
Author
Cajori, Florian, 1859-1930.
Canvas
Page 90
Publication
New York,: The Macmillan company,
1904.
Subject terms
Equations, Theory of
Group theory.

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"An introduction to the modern theory of equations, by Florian Cajori." In the digital collection University of Michigan Historical Math Collection. https://name.umdl.umich.edu/abv2146.0001.001. University of Michigan Library Digital Collections. Accessed May 30, 2025.
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