further mathematics
posted by candice .
The roots of the eqn, x^4 + px^3 + qx^2 + rx + s = 0 where p, q, r, s are constants and s does not equal to 0, are a, b, c, d.
(i) a^2 + b^2 + c^2 + d^2 = p^2 2q (in terms of p & q)
(ii) 1/a + 1/b + 1/c + 1/d = r/s
(in terms of r & s)
(iii) using the above results, or otherwise, express a^3 + b^3 + c^3 + d^3 in terms of p, q & r.
This looks like modern algebra, not high school algebra. I'm not sure how much time I'll devote to this.
You should know that p is sigma1(roots) and q is sigma2(roots). If you know how to work with the symmetric polynomials then the result follows immediately for (i).
If you add the terms in (ii) you get sigma3(roots) / sigma4(roots) and the result follows immediately
For the last part a^3 + b^3 + c^3 + d^3 =
(sigma1)^33(sigma1)(sigma2)+3(sigma3) all of them of the roots
p=sigma1, q=sigma2, r=sigma3 so (iii) is
(p)^33(p)q+3r=a^3+b^3+c^3+d^3
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