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Консультация № 185053
03.01.2012, 22:02
51.94 руб.
0
5
1
Здравствуйте! Прошу помощи в следующем вопросе:Найдите условия на коэффициенты многочлена х^3+ax^2+bx+c, при выполнении которых этот многочлен имеет три различных действительных корня, являющихся последовательными членами некоторой геометрической прогрессии. Заранее благодарен.
Обсуждение
04.01.2012, 01:40
общий
это ответ
Здравствуйте, Тимофеев Алексей Валентинович!
Пусть x1, x2, x3 - три последовательные члены некоторой геометрической прогрессии.
Тогда x2 = x1[$183$]q, x2 = x1[$183$]q2
Если x1, x2, x3 - корни многочлена х3+ax2+bx+c, то
х3+ax2+bx+c = (x - x1)(x - x2)(x - x3) = (x - x1)(x - x1q)(x - x1q2)
х3+ax2+bx+c = x3 - (x1 + x1q + x1q2)x2 + x1q(x1 + x1q + x1q2)x - (x1q)3 = x3 - (x1 + x2 + x3)x2 + x2(x1 + x2 + x3)x - x23
Т.о., получаем:
a = - (x1 + x2 + x3)
b = x2(x1 + x2 + x3) = - a[$183$]x2
c = - x23
Пусть x1, x2, x3 - три последовательные члены некоторой геометрической прогрессии.
Тогда x2 = x1[$183$]q, x2 = x1[$183$]q2
Если x1, x2, x3 - корни многочлена х3+ax2+bx+c, то
х3+ax2+bx+c = (x - x1)(x - x2)(x - x3) = (x - x1)(x - x1q)(x - x1q2)
х3+ax2+bx+c = x3 - (x1 + x1q + x1q2)x2 + x1q(x1 + x1q + x1q2)x - (x1q)3 = x3 - (x1 + x2 + x3)x2 + x2(x1 + x2 + x3)x - x23
Т.о., получаем:
a = - (x1 + x2 + x3)
b = x2(x1 + x2 + x3) = - a[$183$]x2
c = - x23
5
Об авторе:
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен
05.01.2012, 15:30
общий
Адресаты:
Так полученный результат и есть условия на коэффициенты: если они будут связаны указанными равенствами с тремя последовательными членами некой геометрической последовательности, то эти три члена и будут корнями...
Об авторе:
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен
08.01.2012, 10:45
общий
Адресаты:
Формулировка вопроса подразумевает, что условия должны выражаться исключительно в терминах коэффициентов уравнения. Это следующие условия
b+18abc-27c2>0
b3=a3c
|a2-b|[$8805$]2|b|
Эти условия необходимы и достаточны для того, чтобы рассматриваемое кубическое уравнение имело три различных решения, являющихся последовательными членами геометрической прогрессии.
b+18abc-27c2>0
b3=a3c
|a2-b|[$8805$]2|b|
Эти условия необходимы и достаточны для того, чтобы рассматриваемое кубическое уравнение имело три различных решения, являющихся последовательными членами геометрической прогрессии.
Форма ответа
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