### Calculating the first m + 1 partial quotients of the unique positive root of a trinomial ax^{n} + bx + c

We are dealing with polyomial f(x) = ax^{n} + bx + c with integer coefficients, a > 0, c < 0. This will have a unique positive root α which we will assume is greater than 1.
The method of Lagrange (1797) is used to find the the first m + 1 partial quotients of α.
We remark that Disc(f(x)) is given by

Disc(f(x)) = a^{n-2}(γn^{n}c^{n-1}a + β(n-1)^{n-1}b^{n}),

where γ = (-1)^{n(n-1)/2} and β = (-1)^{(n-1)(n-2)/2}.
### References

*Number Theory with Computer Applications*, R. Kumanduri and C. Romero, Prentice Hall 1997, page 261
*Elements of Computer Algebra with Applications*, A.G. Akritas, Wiley 1989, 333-399.
*Art of computer programming*, volume 2, D.E. Knuth, problem 13, Ch. 4.5.3.
*Continued fractions for some algebraic numbers*, S. Lang and H. Trotter, J. für Math. **255** (1972) 112-134; Addendum 267 (1974) ibid. 219-220.
*A new proof of Vincent's theorem*, A. Alesina, M. Galuzzi, L'Enseignement Math **44** (1988), 219-256.
*An explanation of some exotic continued fractions found by Brillart*, H.M. Stark, Computers in number theory, Academic Press 1971, 21-35.

For a program that develops the continued fraction of *all* real roots of a suitably general polynomial, see CALC.

*Last modified 21st July 2006*

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