# Theory of equations by H W Turnbull By H W Turnbull

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For a proof we may refer to We may regard the statement a just geometrical progression summed to infinity or equally well as an example of the binomial l 1 1 expansion of ax~ (l aor )"" Now take \x\ greater than the greatest among |a|, each fraction, and add the results. , |/3 |A|, expand any text on analysis. written as . , (2) the development of R(x] in descending powers of and for valid x, sufficiently large values of \x\. ). , |A|, E(x) is = f(x)/(x) = -Zaa- Unless one or other of a, Therefore if # possible.

II(xa) r quadratic case, when (x) has (f>(x) . x 2 -{-px-}-q, with or without linear The repeated quadratic quadratic factors, and case. In any case the polynomial quotient q(x) must be evaluated by an ordinary division process. In the simple each partial fraction will take the form case, (i), aj(xa) where the denominator is linear and the numerator, being of lower degree, is necessarily a constant Hence we have an identity only. y av ' x-a ' ftx) To evaluate is a alone. On xa = which must therefore be equal to numerator of the partial fractions.

Show from the zigzag that 2x* has a root between 16. and + 8, = x -f 1 1. , digit by digit we need a practical way of expressing /(x) as a polynomial g(y) where y x+a. Equation. +b n _ y+b n g(y) be identically equal when y = xa. Thmf(x) and to are efficients Q l l Q 1 l a t are supposed known to find the coefficients b t ; so too is a. . (I) . (2) The We co- require . , (3) which exhibits/(x) as a polynomial in y of degree n. Hence b n =/(a), 6 n _! = f'(a) and so on. Incidentally this shows that the degree n is the same both for g(y) in y and f(x) in x.