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Консультация № 200921
22.05.2021, 22:52
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Обсуждение
23.05.2021, 00:07
общий
это ответ
Чтобы коэффициенты многочлена были действительными, надо, чтобы каждый комплексный корень имел комплексно сопряженную пару, т.к.
(x-a-b*i)*(x-a+b*i)=( (x-a) - b*i)*( (x-a) + b*i) = (x-a)2 - (b*i))2 = (x-a)2 + b2 = x2-2*a*x+a2+b2
В данном случае для 2 - i следует добавить корень 2 + i, а для 3 - i добавить 3+i.
Таким образом, у нас пять корней (-2, 2 - i, 2 + i, 3 - i, 3+i). Значит, многочлен пятой степени, а именно:
(x +2)*(x - 2 + i)*(x - 2 - i)*(x - 3 + i)*(x - 3 - i)=(x+2)(x2-4*x+5)*(x2-6*x+10) = x5-9*x4+29*x3-31*x2-20*x+50
ОТВЕТ: C*(x5-9*x4+29*x3-31*x2-20*x+50), где C - любое действительное число, отличное от нуля.
Вроде бы так (если не сделал ошибку в перемножении многочленов).
(x-a-b*i)*(x-a+b*i)=( (x-a) - b*i)*( (x-a) + b*i) = (x-a)2 - (b*i))2 = (x-a)2 + b2 = x2-2*a*x+a2+b2
В данном случае для 2 - i следует добавить корень 2 + i, а для 3 - i добавить 3+i.
Таким образом, у нас пять корней (-2, 2 - i, 2 + i, 3 - i, 3+i). Значит, многочлен пятой степени, а именно:
(x +2)*(x - 2 + i)*(x - 2 - i)*(x - 3 + i)*(x - 3 - i)=(x+2)(x2-4*x+5)*(x2-6*x+10) = x5-9*x4+29*x3-31*x2-20*x+50
ОТВЕТ: C*(x5-9*x4+29*x3-31*x2-20*x+50), где C - любое действительное число, отличное от нуля.
Вроде бы так (если не сделал ошибку в перемножении многочленов).
5
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