...whence, the equation a?2 - pas + q = 0, is manifestly equivalent to a?8 - (a + )8) a? + a/3 = 0 : where the coefficient of the second term, with its sign changed, is equal to the sum of the roots, and the third term is equal to their product. 144. Con. 3. Since a? — (a + ft) tv + a/3 = 0, is equivalent...

...sum of the different products of the roots, taken two in a set. 3d. The co-efficient of the fourth term, with its sign changed, is equal to the sum of the different products of the roots, taken three in a set, and so on. 4th. The absolute term, with its...

...another ; let therefore " - _ (í\3 . з з - _ . \3/ ' Hence y3, zs, are the roots of the equation since the coefficient of the second term with its sign changed is equal to their sum, and the last term is equal to their product. Solving this equation, we get / but x = y +...

...curves thus, let ^—(aa: + b)yn~1 + (ca;* + da: + e)yn~*'-' &c. = o Then, since in every equation the coefficient of the second term with its sign changed is equal to the excess by which the sum of the positive exceeds the sum of the negative roots, where this term is deficient,...

...relations, we derive the following conclusions with reference to the values of the co-efficients : 1st. The co-efficient of the second term, with its sign changed, is equal to the algebraic sum of the roots of the equation. 2d. The co-efficient of the third term is equal to the...

...term is equal to the sum of the products of every two of the roots ; the coefficient of the fourth term with its sign changed is equal to the sum of the products of every three of the roots ; and so on. Thus generally if pr denote as usual the coefficient...

...and x — /3 = 0, (х - a) (,r - /8) = 0, Or я* — (a + /3) x + a ß = 0. Whence it appears that the coefficient of the second term, with its sign changed, is equal to the sum of the roots, and the third term is equal to their product. 183. Every equation of the form, x'm + p xm = q ; in...

...relations, we derive the following conclusions with reference to the values of the co-efficients : 1st. The co-efficient of the second term, with its sign changed, is equal to the algebraic sum of the roots of the equation. 2d. The co-efficient of the third term is equal to the...

...arranged according to the descending powers of x and that the coefficient of the first term is 1; then 1. The coefficient of the second term with its sign changed is equal lo the sum of the roots (592, 15, 1C) ; 2. The coefficient of the third term is equal to the sum of...

...term is equal to the sum of the products of every two of the roots ; the coefficient of the fourth term with its sign changed is equal to the sum of the products of every three of the roots ; and so on. Thus generally if pr denote as usual the coefficient...