CHAPTER VI. 



SINGLE EQUATIONS. 



I, Every equation containing one unknown quantity can be 

 put in the form 



Fu nctimi of x=o 



by transposing all the terms to the left side of the equation. 



If it is an equation of the first degree it will always reduce to 

 the form 



.1' — « = o, 



where a must stand for any quantity whatever, positive or nega- 

 tive, integral or fractional, commensurable or incommensurable. 

 It is evident that this equation has the root a and no other. An 

 equation of the first degree might be defined as an equation which 

 can be placed in the form of a rational integral linear function of 

 X equal to zero. 



We have seen that every quadratic equation can be placed in 

 the form 



(x—a)(x—b)^=o, 



which has the two roots a and b and no others. Thus every 

 quadratic equation can be placed in the form of the product of 

 two rational integral linear functions of x equal to zero. 



It will be proved in Part II that every cubic equation can be 

 put in the form 



(x—a)(x — b)(x—c)=o, 



and that it has three roots, a, b, and r, and no others. That is, 

 every cubic equation can be placed in the form of the product ot 

 three rational integral linear functions of x equal to zero. 



It will also be shown that an equation of the fourth degree can 

 be thrown in the form 



(x — a)(x — b)( X — c)(x — d)-=o. 



These and other important properties of equations containing 

 qUC unknown quantity were first discovered by Vieta (1540 — 

 1603), but were independently and more elaborately treated by 

 Harriot (1560 — 1621). 



