Page:Elementary algebra (1896).djvu/487

469 Rh If a, b, c are the roots of x^ -{- qx -{■ r = 0, form the equation whose roots are 19. ka- kb- kc-K 21. ?^+^, ^:^+^, «^^. a^ b^ c- 20. 6%^ c%^ a-^6-^. 22. 6c + ^, ca + ^, a6 + ^.

Solve the equations :

23. 2x4 + x3-^6x2 + a: + 2 = 0. 24. a:4 _ iqjcS _|. 26x'^ - lOx + 1 = 0. 25. x^ - 5 x* + 9 ic^ - 9 a;2 + 5 a: - 1 = 0. 26. 4 a:6 _ 24 a;5 + 57 x* - 73 a:^ + 57 x^ - 24 .r + 4 = 0.

DESCARTES' RULE OF SIGNS.

591. When each term of a series has one of the signs + and - before it, a continuation or permanence occurs when the signs of two successive terms are the same : and a change or variation occurs when the signs of two successive terms are opposite.

592. Descartes' Rule. In any equation, the number of positive roots cannot exceed the number of variations of sign, and in any complete equation the number of negative roots cannot exceed the number of permanences of sign.

Suppose that the signs of the terms in a multinomial are + + — — + — — - + - + - we shall show that if this multinomial is multiplied by a binomial whose signs are + —, there will be at least one more change of sign in the product than in the original multinomial.

Writing only the signs of the terms in the multiplication, we have

+ + - + - - + + - - + - + + - + - + + - + + + - - H - + - + + ±-T + -=F=F+- + - +