Tema1
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Temˇ a
P. 1.1. Folosind identitatea (u + v)3 − 3uv(u + v) − u3 − v 3 = 0 , rezolvat¸i ecuat¸ia redusˇ a de grad 3 x3 + px + q = 0 , efectuˆ and substitut¸ia x = α2 β + αβ 2 ¸si obt¸inˆ and o ecuat¸ie de grad 2 ale cˇarei rˇadˇacini sˇa fie α3 ¸si β 3 . P. 1.2. Rezolvat¸i ecuat¸ia de grad 4 x4 + ax3 + bx2 + cx + d = 0 a efectuˆ and substitut¸ia y = x + a4 , pentru a obt¸ine o ecuat¸ie redusˇ y 4 + py 2 + qy + r = 0. Prin identificarea coeficient¸ilor ˆın y 4 + py 2 + qy + r = (y 2 + αy + β)(y 2 − αy + γ) , exprimat¸i β ¸si γ ˆın funct¸ie de α ¸si obt¸inet¸i o ecuat¸ie de grad 3 ˆın raport cu α2 . O solut¸ie a acestei ecuat¸ii rezolvente conduce la o factorizare care reduce rezolvarea ecuat¸iei reduse la rezolvarea a douˇa ecuat¸ii de grad 2 y 2 + αy + β = 0 ¸si y 2 − αy + γ = 0 .
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Temˇ a
P. 1.1. Folosind identitatea (u + v)3 − 3uv(u + v) − u3 − v 3 = 0 , rezolvat¸i ecuat¸ia redusˇ a de grad 3 x3 + px + q = 0 , efectuˆ and substitut¸ia x = α2 β + αβ 2 ¸si obt¸inˆ and o ecuat¸ie de grad 2 ale cˇarei rˇadˇacini sˇa fie α3 ¸si β 3 . P. 1.2. Rezolvat¸i ecuat¸ia de grad 4 x4 + ax3 + bx2 + cx + d = 0 a efectuˆ and substitut¸ia y = x + a4 , pentru a obt¸ine o ecuat¸ie redusˇ y 4 + py 2 + qy + r = 0. Prin identificarea coeficient¸ilor ˆın y 4 + py 2 + qy + r = (y 2 + αy + β)(y 2 − αy + γ) , exprimat¸i β ¸si γ ˆın funct¸ie de α ¸si obt¸inet¸i o ecuat¸ie de grad 3 ˆın raport cu α2 . O solut¸ie a acestei ecuat¸ii rezolvente conduce la o factorizare care reduce rezolvarea ecuat¸iei reduse la rezolvarea a douˇa ecuat¸ii de grad 2 y 2 + αy + β = 0 ¸si y 2 − αy + γ = 0 .
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