1 Abstract 2 3 n a ax 2 + bx + c = 0 (a 0) (1) ( x + b ) 2 = b2 4ac 2a 4a 2 D = b 2 4ac > 0 (1) 2 D = 0 D < 0 x + b 2a = ± b2 4ac 2a b ± b 2

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1 1 Abstract n a ax + bx + c = 0 (a 0) (1) ( x + b ) = b 4ac a 4a D = b 4ac > 0 (1) D = 0 D < 0 x + b a = ± b 4ac a b ± b 4ac a b a b ± 4ac b i a D (1) ax + bx + c D 0 () ()

2 ( ) 1. D = b 4ac () 4 x 1,..., x n d i c i1...inx x i n n i 1 + +in=d i 1,...,in N 0 b 4ac a, b, c (1) α, β ax + bx + c = a(x α)(x β) = a(x (α + β)x + αβ) α + β = b a, αβ = c a = α β 1 a =(α β) = α + β αβ = (α + β) 4αβ ( ) b = 4 c a a = b 4ac a D = a = a (α β) = b 4ac () 4 D () (i) D > 0 (ii) D = 0 (iii) D < 0 ((i)(iii) = (ii) (i)(iii) = α α )

3 ( ) 1. 1 f(x) f(x) = 0 { f(x) = 0 f (x) = 0 f(x) = 0 α g(x) f(x) = (x α) g(x) f (x) = (x α)g(x) + (x α) g (x) f (α) = 0 f(α) = 0, f (α) = 0 h(x) f(x) = (x α)h(x) f (α) = 0 f (x) = h(x) + (x α)h (x) h(α) = 0 k(x) h(x) = (x α)k(x) f(x) = (x α) k(x) α f(x) (1) ax + bx + c = 0 (4) ax + b = 0 (5) (4) x (5) b (5) a (6) bx + c = 0 (6) b 4ac = 0 (7) (7) 1 a (6) b (5) (7)

4 ( ) 4.1 ax + bx + cx + d = 0 (a 0) (8) y = x + b a () x = y b/(a) (8) ) ( ay + ( b b a + c y + 7a bc ) a + d = 0 a p = b a + c a = b + ac a y = u + v (9) q = b 7a bc a + d a = b 9abc + 7a d 7a y + py + q = 0 (9) u + v + (uv + p)(u + v) + q = 0 { u + v + q = 0 uv + p = 0 u, v y = u + v (9) u, v u + v = q, ( p u v = ) ( p ) t + qt = 0 ( ) t = q ± ( q ) + ( p ) u, v t u uv = p v ( q ) ( p R = + )

5 ( ) 5 q + R q ( q ) ( R = p ) p R = = ω = ( 1 + i)/ (9) q y 1 = + q R + R q y =ω + q R + ω R q y =ω + q R + ω R ( []) ( q ) ( p b R = + = ) c + 7a d 18abcd + 4b d + 4ac 108a 4 D = 108a 4 R = b c 7a d + 18abcd 4b d 4ac (10) D = 7a q = b + 9abc 7a d (11) q + R = D 7a + D a 4 = 1 a D + a 7D (8) α 1 = 1 a α = 1 a α = 1 a ( b + D + a ) D 7D + a 7D ( b + ω D + a ) 7D + ω D a 7D ( b + ω D + a ) D 7D + ω a 7D (8) D, D (10)(11) (i) (ii) (iii) (iv) D > 0 D = 0, D 0 1 D = D = 0 D < 0 1 (i) D > 0 D + a D 7D, a 7D

6 ( ) 6 α 1 ω D + a 7D = ω D a 7D α, α (ii) D = 0, D 0 α 1 α = α (iii) D = D = 0 (iv) D < 0 α 1 α, α (8) D = b c 7a d + 18abcd 4b d 4ac 1 x 7x 6 = (x )(x + 1)(x + ) = 0 p = 7, q = 6 ( q ) ( p R = + ) ( ) 6 = + ( ) 7 = = = 10 9 < i = i i = (a + b i) a, b ( ) { a 9ab = 81 a b b = 10 b = ±, ±5 a =, b = x 7x 6 = 0 ( α 1 = 1 + ) i + ( α =ω 1 + ) i 1 81 ± 0 i = 1 ± i ( α =ω 1 + ) i + ω ( 1 ) i = ( ) + ω i 1 ( 1 ) i = = 1 ()

7 ( ) 7 (00) x + px + q = 0( p, q ) p < q < p ( () 00 ) < 0. (8) α, β, γ ax + bx + cx + d = a(x α)(x β)(x γ) =a(x (α + β + γ)x + (αβ + βγ + γα)x αβγ) α + β + γ = b a, αβ + βγ + γα = c a, αβγ = d a =(α β) (α γ) (β γ) =(αβ + βγ + γα) (α + β + γ) 7α β γ + 18αβγ(α + β + γ)(αβ + βγ + γα) 4(αβ + βγ + γα) 4αβγ(α + β + γ) Maple α, β, γ a,b,c A := a*b*c; a b c B := a*b+a*c+b*c; a b + a c + b c C := a+b+c; a + b + c DD := B^*C^-7*A^+18*A*B*C-4*B^-4*A*C^; (a b + a c + b c) (a + b + c) - 7 a b c + 18 a b c (a b + a c + b c) (a + b + c) - 4 (a b + a c + b c) -4 (a + b + c) a b c

8 ( ) 8 ( ) DD := expand(dd); a b - a b c + a c - a b + a b c + a b c a c + a b + a b c - 6 a b c + a b c + a c a b c + a b c + a b c - a b c + b c 4 - b c + b c DD := factor(dd); (b - c) (a - c) (a - b) = 1 ( b a 4 c 7a d + 18abcd 4b d 4ac ) a 4 D = a 4 = b c 7a d + 18abcd 4b d 4ac (1) x 1,..., x n f(x 1,..., x n ) x 1,..., x n x i, x j f f s 1 = n n n x i, s = x i x j, s = x i x j x k,, s n = 1 i<j n 1 i<j<k n n x i x 1,..., x n f(x) = a n x n + a n 1 x n a 1 x + a 0 a i a n = ( 1) n i s n i, i = 0,..., n 1

9 ( ) 9 (p.140, p.9 ). (1) f(x) = a n x n + a n 1 x n a 1 x + a 0 α 1,..., α n = (α 1,..., α n ) = f(x) (discriminant) D = D(f) = a n n = an n 1 i<j n (α i α j ) 1 i<j n (α i α j ) (1) 1. 1 (1) α 1,..., α n (n ) a n n a n n () ( 1) ax + bx + c (α i α j ) a (α 1 α ) = a n n 1 i<j n b 4ac (α i α j ) b 4ac 1 i<j n a 4. f(x) = a n x n + a n 1 x n a 1 x + a 0 = a n g(x) = b n x n + b n 1 x n b 1 x + b 0 = b m n m j=1 (x α i ) (x β j )

10 ( ) 10 (resultant) n m R(f, g) = a m n b n m (α i β j ) j=1 4 f(x), g(x) ( ) R(f, g) = 0 1 n R(f, g) = a m n g(α i ) = ( 1) mn b n m n m n R(f, g) = a m n b m (α i β j ) = a m n g(α i ) j=1 ( m R(f, g) = b n m a m n ( 1) n n m f(β j ) j=1 ) m (β j α i ) = ( 1) mn b n m f(β j ) j=1 j=1 5 f(x) = a n x n + a n 1 x n a 1 x + a 0 f(x) α 1,..., α n 1 R(f, f ) = a n 1 n f (x) = a n n n D(f) = ( 1) n(n 1)/ 1 a n R(f, f ) f(x) = a n n (x α i ) (x α 1 ) (x α i 1 )(x α i+1 ) (x α n ) f (α i ) = a n 1 n a n n = ( 1) n(n 1)/ a n a n n n 1 i<j n n (α i α j ) j=1 j i (α i α j ) = ( 1) n(n 1)/ a n D(f)

11 ( ) 11 6 ( [6]) (n + m) R(f, g) = a n a n 1 a 1 a a n a n 1 a a n a n 1 a 0 b m b m 1 b 1 b b m b m 1 b 1 b b m b m 1 b 0 Sylvester R Sylvester (f, g) (pp ) 7 a 0 a 1 a n 1 a n a 0 a 1 a n 0 R Seki (f, g) = 0 0 a 0 a 1 a n b 0 b 1 b m 1 b m b 0 b 1 b m 1 b m b 0 b 1 b m =( 1) mn R Sylvester (f, g) m = n 1 nm 0 (mod ) R Seki (f, g) = R Sylvester (f, g) 4 (...) () (...) [] ax + bxy + cy, a, b, c Z

12 ( ) 1 (a, b, c) (a, b, c) (substitutiones) x = αx + βy, y = γx + δy (a, b, c ) a x + b x y + c y =a(αx + βy ) + b(αx + βy )(γx + δy ) + c(γx + δy ) =(aα + bαγ + cγ )x + (aαβ + b(αδ + βγ) + cγδ)x y + (aβ + bβδ + cδ )y a =aα + bαγ + cγ b =aαβ + b(αδ + βγ) + cγδ c =aβ + bβδ + cδ b a c =(aαβ + b(αδ + βγ) + cγδ) (aα + bαγ + cγ )(aβ + bβδ + cδ ) =(b ac)α δ (b ac)αβγδ + (b ac)β γ =(b ac)(αδ βγ) b ac (αδ βγ) = 1 (a, b, c) bb ac (a, b, c) (determinantem) [4, p.17] (, ) {( α β GL (Z) = γ δ ) } αδ βγ = ±1; α, β, γ, δ Z GL (Z) (unimodular matrix) {( α β SL (Z) = γ δ GL (Z) ) } αδ βγ = 1; α, β, γ, δ Z GL (Z) 19

13 ( ) 1 4. t C t : f(x, y, t) = 0 (14) E (14) E {C t } E C t (ϕ(t), ψ(t)) E x = ϕ(t), y = ψ(t) (15) (14) E (ϕ(t), ψ(t)) C t E f x (ϕ(t), ψ(t), t)ϕ (t) + f y (ϕ(t), ψ(t), t)ψ (t) = 0 f(ϕ(t), ψ(t), t) = 0 (16) t f x (ϕ(t), ψ(t), t)ϕ (t) + f y (ϕ(t), ψ(t), t)ψ (t) + f t (ϕ(t), ψ(t), t) = 0 f t (ϕ(t), ψ(t), t) = 0 (17) E {C t } (16) (17) (16) (17) t R(f, f t ) = 0 f(x, y, t) = 0 8 C t E E R(f, f t ) = 0 R(f, f t ) = 0 E C t f x (x, y, t) = f y (x, y, t) = 0 D R f : D R f(x, y) (a, b) U y f y (x, y) f(x, y) f(a, b) = 0, f y (a, b) 0 x = a V V y = g(x) b = g(a), f(x, g(x)) 0

14 ( ) 14 f(x, y) U C 1 g(x) C 1 ([10] ) g (x) = f x(x, y) f y (x, y)) I, U R, R Φ : I R, f : U R f(i) U Φ(t) t f(x, y) Φ(t) = (ϕ(t), ψ(t), χ(t)) g(t) = f(φ(t)) = f(ϕ(t), ψ(t), χ(t)) t g (t) = f x (ϕ(t), ψ(t), χ(t))ϕ (t) + f y (ϕ(t), ψ(t), χ(t))ψ (t) + f z (ϕ(t), ψ(t), χ(t))χ (t) (015) a C : y = ax + 1 4a 4a a C y = ax + 1 4a 4a = (x 1)a + 1 4a x > 1 y = (x 1)a + 1 4a (x 1)a 1 4a = x 1 y = x 1 x = ±1 y = 1 4a y > 0 x < 1 lim y = +, a +0 C lim y = a {(x, y) y x 1 x > 1} { (x, y) x < 1 } { (x, y) x = ±1 y > 0 } y > 0 y 0 ( Mac Grapher )

(015 8 1 ) 15 1-5 -4 - - -1 0 1 4 5-1 - - f(x, y, a) = ax + 1 4a 4a y a f(x, y, a) = 0 : (x 1)a + 1 4a y = 0 f a (x, y, a) = 0 : x 1 4a 1 = 0 y = x 1 C () y 5 19 (168) { f(x) = a0 + a 1 x + a x + + a

15 ( ) f(x, y, a) = ax + 1 4a 4a y a f(x, y, a) = 0 : (x 1)a + 1 4a y = 0 f a (x, y, a) = 0 : x 1 4a 1 = 0 y = x 1 C () y 5 19 (168) { f(x) = a0 + a 1 x + a x + + a n x n = 0 g(x) = b 0 + b 1 x + b x + + b m x m = 0 R Seki (f, g) = ( 1) mn R Sylvester (f, g) = ( 1) mn R(f, g) x a 0,, a n, b 0., b m R Seki (f, g) = 0 ([5]1685) f(x) = a 0 + a 1 x + a x + + a n x n = 0

16 ( ) 16 ( ) 1 n n a n R Seki (nf xf, f ) = ( 1) n(n 1)/ D(f) f(x) = a 0 + a 1 x + a x + a x = 0 D = 0 R Seki (f, 1! x f ) = 1 8a R Seki (f, f ) = 0 [14] [] [1] 1841 Q = Ax + Bx y + Cxy + Dy x, y Q x = 0, Q y = 0 θ(q) =(ad bc) 4(b ac)(c bd) =a d b c + 4ac + 4b d 6abcd [7] 1851 s 0 x + s 1 x y + s xy + s y I = 1ϵ(s 1s 7s 0s + 18s 0 s 1 s s 4s 1s 4s 0 s ) = 1ϵD I (discriminant) [1] George Boole, Exposition of a general theory of linear transformations, Part I., The Cambridge mathematical Journal Vol. III. No.XIII (1841), 1-0.

17 ( ) 17 [] Gerolamo Cardano, Ars Magna, English translation The Rules of Algebra: (Ars Magna) (Dover Books on Mathematics) [] Friderico Gauss, Disquisitiones Arithmeticae, 1801, in Carl Friedrich Gauss Werke, Erster Band, 186. [4] 1995 [5] 16-8 [6] J.J. Sylvester, A method of determining by mere inspection the derivatives from two equations of any degree, Philosophical Magazine, XVI. (1840), 1-15, in The Collected Mathematical Papers of James Joseph Sylvester, vol. I, [7] J.J. Sylvester, On a remarkable discovery in the theory of canonical forms and of hyperdeterminants, Philosophical Magazine, II. (1851), , in The Collected Mathematical Papers of James Joseph Sylvester, vol. I, 65-8 [8] J I 1985 [9] 198( 198) [10] ( ) 198 [11] Victor J. Katz, A History of Mathematics, rd ed., Addison-Wesley, 009. [1] ([11] ) 005 [1] [14] [15] 1974 [16] 1965

ax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 4

ax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 4 20 20.0 ( ) 8 y = ax 2 + bx + c 443 ax 2 + bx + c = 0 20.1 20.1.1 n 8 (n ) a n x n + a n 1 x n 1 + + a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 444 ( a, b, c, d