IMPOSSIBLE roots, in algebra. To discover how many impossible roots are contained in any proposed equation, Sir Isaac Newton gave this rule in his alge bra, viz. Constitute a series of fractions, whose denominators are the series of na tural numbers, 1, 2, 3, 4, 5, &c. continued to the number showing the index or expo nent of the highest term of the equations, and their numerators the same series of numbers in the contrary order ; and di vide each of these fractions by that next before it, and place the resulting quo tients over the intermediate terms of the equation ; then, under each of the inter mediate terms, if its square multiplied by the fraction over it be greater than the product of the terms on each side of it, place the sign + ; but if not, the sign ; and under the first and last term place the sign Then will the equa tion have as many imaginary roots as there are changes of the underwritten signs from -1- to , and from to So for the equation x3 4x' -I- 4x 6 = 0, the series of fractions is 44,4: then the pecond, divided by the first, gives, a or 4, and the third divided by the second gives 4. also ; hence these quotients, placed over the intermediate terms, the whole will stand thus : 7 7 + Now because the square of the second term, multiplied by its superscribed frac tion, is ' x4, which is greater than 4 ac5, the product of the two adjacent terms, therefore the sign is set below the se cond term ; and because the square of the third term, multiplied by its over written fraction, is y x', which is less than 24x', the product of the terms on each side of it ; therefore the sign is placed under that term ; also the sign + is set under the first and last terms. Hence
the two changes of the underwritten signs -I- -F, the one from + to , and the other from to show that the given equation has two impossible roots. When two or more terms are wanting together, under the place of the first of the deficient terms write the sign un der the second the sign +, under the third , and so on, always varying the signs, except that under the last of the deficient terms must always be set the sign +, when the adjacent terms on both sides of the deficient terms have contrary signs. As in the equation, ±a5=0, which has four imaginary roots.