By K. F. Riley, M. P. Hobson, S. J. Bence

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**Sample text**

B0 − αb1 = a1 , −αb0 = a0 . These can be solved successively for the bj , starting either from the top or from the bottom of the series. In either case the ﬁnal equation used serves as a check; if it is not satisﬁed, at least one mistake has been made in the computation – or α is not a zero of f(x) = 0. We now illustrate this procedure with a worked example. Determine by inspection the simple roots of the equation f(x) = 3x4 − x3 − 10x2 − 2x + 4 = 0 and hence, by factorisation, ﬁnd the rest of its roots.

Roots that are complex (see chapter 3) do not have such a graphical interpretation. For polynomial equations containing powers of x greater than x4 general methods do not exist for obtaining explicit expressions for the roots αk . Even for n = 3 and n = 4 the prescriptions for obtaining the roots are suﬃciently complicated that it is usually preferable to obtain exact or approximate values by other methods. Only for n = 1 and n = 2 can closed-form solutions be given. These results will be well known to the reader, but they are given here for the sake of completeness.

However, at this stage we will draw our illustrative and test examples from earlier sections of this chapter and other topics in elementary algebra and number theory. 1 have already shown the way in which an inductive proof is carried through. They also indicated the main limitation of the method, namely that only an initially supposed result can be proved. Thus the method of induction is of no use for deducing a previously unknown result; a putative equation or result has to be arrived at by some other means, usually by noticing patterns or by trial and error using simple values of the variables involved.