Roots of Cubic Equations
Date: 01/13/2002 at 00:03:55 From: John Devlin Subject: Roots of Cubic Equations Can you help me with this? a, b, and c are the roots of the equation x^3 - px^2 + qx - r = 0. Express the following in terms of p, q, and r. 1) a^2 + b^2 + c^2 2) 1/a^2 + 1/b^2 + 1/c^2 Thank you very much.
Date: 01/15/2002 at 17:04:28 From: Doctor Schwa Subject: Re: Roots of Cubic Equations >1) a^2 + b^2 + c^2 Hint: a^2 + b^2 + c^2 = (a+b+c)^2 - 2(ab + ac + bc). Does that help? >2) 1/a^2 + 1/b^2 + 1/c^2 Hint: If a is a root of the given polynomial, then 1/a is a root of (1/x)^3 - p(1/x)^2 + q(1/x) - r, and then multiplying by x^3, 1/a is also a root of -rx^3 + qx^2 -px + 1, and then dividing through by -r will put it in the same pattern as your problem (1). I hope that helps. These are neat problems, the kind I'd expect to see on a math contest more than in the typical math class. Feel free to write back if you'd like another hint. - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/
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