Similarly qn-1 is the remainder found by dividing the last expression by x — h, and the quotient arising from the division is
qo^x - hf -' + q,(:x - h)-^ + • • • + qn-^2 ;
and so on. Thus qn, qn-1, qn-2 may be readily found by Synthetic Division. The last quotient is q0 is obviously equal to p0.
Hence to obtain the transformed equation,
Divide f(x) by x±h according as the roots are to be greater or less by h than those of the original equation, and the remainder will be the last term of the required equation. Divide the quotient thus found by x ± h, and the remainder will be the coefficient of the last term but one of the required equation; and so on.
Ex. Find the equation whose roots exceed by 2 the roots of the equation 4x4 + 32x3 + 83x2+76x4-21 =0.
The required equation will be obtained by substituting x — 2 for x in the proposed equation ; hence in Horner's process we employ x + 2 as divisor, and the calculation is performed as follows :
+ 2 4 32 83 76 21 4 24 35 6 |9 4 16 3 |0 4 8 1 -13 4 |0 4
Thus the transformed equation is
4x4 - 13x2 + 9 = 0, or (4x2 - 9)(x2 - 1)= 0.
The roots of this equation are + f , — f , +1, — 1 ; hence the roots of the given equation are -1, -I, -1, -3.
585. To transform a complete equation into another which wants an assigned term.
The chief use of the substitution in the preceding article is to remove some assigned term from an equation.
