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3. Transform the equation
x12x +17x49x+7= 0, into another which shall not contain the 3d power of the unknown quantity.
12 By (437), put x =y+;
= 3+y. Here x' = 3 and m= = 4. X'= (3)412(3)+17(3)—9(3)+7, or X =-110. X', = 4(3)36(3) +34(3)-9,
X = -123. X2
X" = 6(3):—36(3)+17,
X' = 4(3)%12,
2:3 Therefore the transformed equations must be
y-37y'--123y-110 = 0. 4. Transform the equation
--6x +13x-12 = 0, into another wanting its second term.
2+y; then X = (2)-6(2)+13(2)-12,
-2. X', = 3(2)-12(2)+13,
2.3 Therefore, the transformed equation must be
yo+y^2 = 0. 5. Transform the equation
x*--4x' _8x+32 = 0, into another whose roots shall be less by 2. Put x = 2+y.
Ans. y' +4yo_24y = 0. As this transformed equation has no term independent of y, y = 0 is one of its roots, and x = 2 is therefore a root of the original equation.
Put x =
6. Transform the equation,
** +16x*+99x2 +228x+144 = 0, into another whose roots shall be greater by 3.
Ans. y* +4y8 +9y-42y = 0. 7. Transform the equation,
2* —8x8 +ä? +82x—60 = 0, into one incomplete in respect to its second term.
Ans. y' -23y*+22y+60 = 0. 439. Resuming the transformed equation (1'), (436), which is X.
y +...+ yo+**/*+x,+X= 0, 1.2...(m-1)
2:3 and replacing y by its value, y = x-', it becomes
= 0. X
2 Now it is evident that, by developing the first member of this equation and arranging the result with reference to the descending powers of x, the first member of the original equation will be reproduced; for, by this operation we will have merely retraced the steps by which eq. (1') was derived from eq. (1) in the article referred to Hence we have the identical equation, *"+ A3-!+B.cm-2+.... S.x*+To+U
2 The quotients and remainders obtained by the division of the first member of this equation by any quantity, will not differ from those arising from the division of the second member by the same quantity. Dividing the second member by x—X', the first remainder is X', and the quotient,
X'. (x-2')-1. 1.2..(m-1)
X + 2.3
and this divided again by x—x', will give for the second remainder X", and the quotient, X'
2:3 It is unnecessary to continue this process further, to see that these successive remainders are the coefficients of the transformed equation (1') beginning with the absolute term, or the coefficient of yo. The divisor to be employed is x—x' if the roots of the transformed equation are to be less, in value, than those of the given equation by the constant difference x'; if greater, the divisor must be r+x'. Hence, an equation may be transformed into another of which the roots are greater, or less, than of the given equation by by the following
RULE. I. Divide the first member of the given equation (the seconil member being zero) by x plus the constant difference between the roots of the two equations, continuing the operation until a remainder is obtained which is independent of x; then divide the quotient of this division by the same divisor, and so on, until m divisions have been performed.
II. Write the transformed equation, making these successive remaindlers the coefficients of the different powers of the unknown quantity, beginning with the zero power.
It must be borne in mind that the term plus in this rule is used in its algebraic sense.
By a little reflection, it will seem that the mth quotient will be the coefficient of am in the original equation, and that this will also be the coefficient of the highest power of the unknown quantity in the transformed equation.
1. Transform the equation,
x*_4x48x+32=0, into another of which the roots shall be less by 2. This is example 5 of the last article. Make
x = 2+y, or y=3 -2;
then the operation is as follows:
-2:0$ _ 8x -2x+42
-4x*—83 x-2)2--20-4-16(2_4 -4x+82
-40-16 -16x +32
-4x+ 8 0 = X
-24 = X'
1.2 Hence, the transformed equation is
y* +4y° +0y2—24y+0=0; or,
y'+4y8–24y=0, as before. 2. Transforın the equation, x*—12x+16x*_9x+7 = 0, into one having roots less by 3.
Here x=y+3, or y = x-3.
-110 = X', 1st remainder:
-123 = X',, 2d remainder. 2-32-60-28(x-3 x-3):43(1 X_3x
X", m -3.0+ 9
Hence, y*+0y' — 37y* — 123y — 110 = 0, is the transformed equation.
We shall have 4 remainders, if we operate on an equation of the 4th degree; 5 remainders with an equation of the 5th degree; and, in general, n remainders with an equation of the nth degree.
The transformation of equations by division, treated of in this article, if performed by the ordinary rule, would be too laborious for practical application ; but by a modified method of division, called Synthetic Division, it becomes expeditious and easy.
As preliminary to the explanation of this method of division, we must explain the process of
MULTIPLICATION AND DIVISION BY DETACHED
440. It has been seen that when two polynomials are homogeneous their product is also homogeneous, and the number which denotes its degree is the sum of the numbers denoting the degrees of the factors. It is evident that if the polynomials contain but two letters, and both are arranged with reference to the same letter, the product will be arranged with reference to that letter. Since, in the operation of multiplying the terms of the multiplicand by the terms of the multiplier, the products of the coefficients are not