EQUATION. 



to n factors. For by Actually multiply- 

 ing the factors together, we obtain a 

 quantity of n dimensions similar to the 

 proposed quantity z n ~p z l -j- q z *, 

 &c. ; and if a, 6, c, &c. can be so assum- 

 ed, that the coefficients of the corres- 

 ponding terms in the two quantities be- 

 come equal, the whole expressions coin- 

 cide. And these coefficients may be made 

 equal, because these will be n equations, 

 to determine n quantities, a, b, c, &c. If 

 then the quantities, a, b, c, &c. be pro- 

 perly assumed, the equation z p z" '+ 

 q zjtt-V&c. = 0, is the same with z a 

 X z b X z c, &c. = 0. The quanti- 

 ties a, b y c, d, &c. are called roots of the 

 equation, or values of z ; because, if 

 any one of them be substituted for z, the 

 whole expression becomes nothing, which 

 is the condition proposed by the equa- 

 tion. 



Every equation has as many roots as it 

 has dimensions. If zn p z- 1 +/> z**-% 

 &c. = 0, or z a X 2 b X * c, &c. 

 to n factors = 0, there are n quantities, 

 , b, c, &c. each of which when substi- 

 tuted for z makes the whole =** 0, because 

 in each case one of the factors bee omes 

 = ; but any given quantity different 

 from these, as e when substituted for 0, 

 gives the product e a X e b X? c, 

 &c. which does not vanish, because none 

 of the factors vanish, that is, e will not 

 answer the condition which the equation 

 requires. 



When one of the roots, a, is obtained, 

 the equation z aX z b X z c, &c. 

 = 0, znp z"- 1 -f- q Z" 1 , &c. = is di- 

 visible by z a without a remainder 

 and is thus reducible to z b X z c, 

 8tc. 0, an equation one dimension low- 

 er, whose roots are b and c. 



Ex. One root of #3 -f- 1 = 0, or x + 

 1 = 0, and the equation may be de- 

 pressed to a quadratic in the following 



-f- 1 )a:3+l (or 1 x+l 



Hence the other two roots are the roots 

 of the quadratic x 1 x + 1 = 0. If 

 two roots, a and 6, be obtained, the equa- 

 tion is divisible by x a X x 6, and 

 may be reduced in the same manner two- 

 dimensions lower. 



Ex. Two roots of the equation z 6 1 

 = 0, are -f 1 and 1, or z 1 = 0, and 

 z -h 1 = ; therefore it may be depress- 

 ed to a biquadratic by dividing by z 1 



Hence the equation z+ -f- z* -\- 1 =0 con- 

 tains the other four roots of the proposed 

 equation. 



Conversely, if the equation be divisi- 

 ble by a: a without a remainder, a is 

 a root; ifbyx a X x 6, a and b are 

 both roots. Let Q, be the quotient aris- 

 ing from the division, then the equation 

 is x a Xx b x Q = 0, in which, if 

 a or b be substituted for x, the whole 

 vanishes. 



EQUATIONS, cubic solution of, by Car- 

 dan's rule. Let the equation be reduced 

 to the form x3 q a;+r=p, where q and 

 r may be positive or negative. 



Assume x = a -{- b, then the equation 

 becomes'a -j- 6\3 g X a + b -f r=0, or 



b 



? x 



r = ; and since we have two unknown 

 quantities, a and 6, and have made only 

 one supposition respecting them, viz. that 

 a -\- b = x, we are at liberty to make 

 another ; let 3 a b q =0, then the 

 equation becomes a?-\-b3 + r = ; also, 



since 3ab ?=0, b =^- , and by sub- 

 3a 



stitution, 



27 



+ r = 0, or a 6 



r a3 -{- o^- = 0, an equation of a qua- 

 dratic form; and by completing the 



