Resolution of Algebraic Equations. 297 
that remain ; and here again, the cube produced will be real. 
But, if all the roots are real, and unequal, their sums and diffe- 
rences will all be real : whence, all the binomials will involve 
the imaginary surd ; which constitutes the irreducible case. 
To give examples of this, let, 1st. (a; 1 — 21 -f 4 = o), a 
cubic equation, whose roots are (2, — t -\- </ — 1, — 1 — — ■ 1); 
the binomials constructed by taking their sums and differences 
as before, will be 
2 — i + v/ — 1 = 1 -+-</ — 1 
2 + 1 — v/ — 1 = 3 — \/ — 1 
2—1 — v/ — 1 = 1 — v/ — 1 
2 + 1 + v/— 1 = 3 + \? — 1 
1 1 — v — 1 — - \ which last bi- 
— i+v/- l + i + v/ — 1= 2 v/-iJ 
nomial — 2+2 / — 1|, when the latter quantity (2/ — 1) is 
drawn into the imaginary surd ( v/ — j-)> becomes j — 2 — 
a real quantity. 
2dly. Let (x s — 3 .z -f- 2 = o) be proposed, whose roots 
have been, in Art. 23, shewn to be (2, — 1, — 1). Here 
2 “l = + ll 
2 + 1 = + 3 J 
2-1 = + !} 
2 + 1 = + 3 J 
1 - - - 1 — ^ 1 
V. This latter binomial must evidently remain 
+ oj J 
real, since the difference into which the imaginary factor was 
to have been drawn vanishes. 
