An introduction to the modern theory of equations, by Florian Cajori.

36

THEORY OF EQUATIONS

Ex. 1. Transform x4 - 5 x3 + 7 x2 - 4 x + 5 -= 0 into another equation
whose roots are less by 2.
By synthetic division the process is as follows:
1 -5 +7 -4 +5L 2
+2 -0 +2 — 4
-3 +1 -2 +1
+2 -2 -2
-1 -1 -4
+2 +2
+1 +1
+2
+3
The numbers in black type, 1, - 4, + 1, + 3, indicate, respectively, the
first, second, third, and fourth remainder. Hence the required equation
is X4 + 3x3 + x2 -4 x + 1 = 0.
Ex. 2. Diminish the roots of 2 x - xs + 10 x - 8 = by 5.
Ex. 3. Transform the equation x4 - 8 x-3 + x2 + - 6 0 into another
in which the second term is wanting.
The sum of the roots of the given equation, by ~ 13, is + 8. In the
required equation the sum shall be zero.' Hence the sum of the roots
must be diminished by 8; each single root by 2. Hence we get by
synthetic division x4 - 23 Xs - 59 x - 48 0.
Ex. 4. Remove the second term of x5 + 10 r4 + x2 + 1 = 0.
Ex. 5. Remove the second term of 4 X4 + 8 x~ + x + 12 -0.
33. Removal of Second Term in the Cubic. In the transformation of the general cubic
b0x + 3 bj' + 3 b, + b3 = O
into another, deprived of the second term, we notice that each
root must be increased by,- the sum of the roots in the given
3 bb, the bl
3b,
cubic being - - Put y = x +,,then x =? b Substitutbo bO' bo
ing, we obtain
bo - )l 3 ( ) + 6b b2 b - = 0
\ IO/ \ ^(Vj\ ^O/