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But equations (2) and (3) are identical, for the sum of the positive terms in each is equal to the sum of the negative terms, and therefore they are identical. Now if a be a root of equation (1), and if a be substituted for x in equation (1) and—a in equation (2), if ʼn be an even number, or in equation (3) if n be an odd number, the results will be the very same; and since the former is verified by such substitution, a being a root, the latter, viz., equation (2) or (3), as the case may be, is also verified, and therefore a is a root of the identical equations (2) and (3).

Corollary. If the signs of all the terms are changed, the signs of the roots remain unchanged.

EXAMPLES.

(1) The roots of the equation 23-6x2+11x-6=0 are 1, 2, 3. What are the roots of the equation x3+6x2+11x+6=0?

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(2) The roots of the equation -6.x3+24x-16=0 are 2, −2, 3± √5. Express the equation whose roots are 2, −2, −3+ √5, and −3— √5. Ans. x+6x-24x-16=0.

PROPOSITION VIII.

248. Surds and impossible roots enter equations by pairs.

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Let x+A+A.2.2a¬2 + · · · · An-1+A=0 be an equation having a root of the form a+b√-1, then will a-b√-1 be also a root of the equation; for, let a+b 1 be substituted for r in the equation, and we have

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(a+b √ −1)" + A1(a+b √ −1 )"~1+........ A„-1(a+b√ −1)+A„=0. Now, by expanding the several terms of this equation, we shall have a series of monomials, all of which will be real except the odd powers of b√ −1, which will be imaginary. Let P represent the real and Q√1 the imaginary terms of the expanded equation; then

P+QV-1=0,

an equation which can exist only when P=0 and Q=0, for the imaginary quantities can not cancel the real ones, but the real must cancel one another, and the imaginary one another separately.

Again, let a—b √-1 be substituted for x in the proposed equation; then the only difference in the expanded result will be in the signs of the odd powers of b√-1, and the collected monomials, by the previous notation, will assume the form P-Q√-1 but we have seen that P=0 and Q=0;

.. P-Q√-1=0,

and hence a-b√-1 also verifies the equation, and is therefore a root. Such roots are called conjugate.

In a similar manner, it is proved that if a+ √b be one root of an equation, a—√b will also be a root of that equation.

Corollary 1.-An equation which has impossible roots is divisible by

{x—(a+b√−1)}{x—(a—b √ −1)}, or x2—2ax+aa+b2,

and, therefore, every equation may be resolved into rational factors, simple or quadratic.

Corollary 2.—All the roots of an equation of an even degree may be impos

sible, but if they are not all impossible, the equation must have at least two real roots.

Corollary 3.-The product of every pair of impossible roots being of the form ab is positive; and, therefore, the absolute term of an equation whose roots are all impossible must be positive.

Corollary 4.-Every equation of an odd degree has at least one real root, and if there be but one, that root must necessarily have a contrary sign to that of the last term.

Corollary 5.-Every equation of an even degree whose last term is negative has at least two real roots, and if there be but two, the one is positive, and the other negative.

PROPOSITION IX.

249. The m roots of the equation X=0, or

x+Px-1+QxTM¬2+, &c, =0....

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must be of the form a+b √ −1, of which form we have already shown (Art. 241) that it must have one.

We

For, let a+b√-1 be the root whose existence is demonstrated. know (Prop. II.) that the polynomial +, &c., is divisible by x-(a+b√−1); but when we effect this division, the quantities a+b √ −1, P, Q, &c., can combine only by addition, by subtraction, and by multiplication; then the coefficients of the quotient xm-1+, &c., will still be of the form a+b√ −1. Consequently, the equation am-1+, &c., will also have at least one root of the form a'+b'√—1; dividing am-1+, &c., by x—(a'+b' √ −1), the coefficients of the quotient xm-2+, &c., will be still of the same form. Continuing to reason thus, it is evident that the primitive polynomial X will be divided into m factors of the form r−(a+b√ −1), and, consequently, the roots of the equation will all be of the form a+b√-1.

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Let x=a+b√-1 be a root of equation (1), and Y'+Z'√-1 the quotient of its first member, by x-a-b√1, we have the identity

(Y'+Z' √ −1)(x—a—b √ −1)=Y+Z√−1..

Effecting the mu

lication in the 1° member, we find

(x − a)Y'+bZ'+[(.x—a)Z' —bY] √ −1

.

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(3)

Changing now in the two factors Z' into -Z', and b into b, we see that in the product the part which does not contain v −1 remains the same, and that that which does contain -1 only changes its sign; by virtue of (3), therefore, we have

(Y'-Z'√1)(x-a+b√-1)=Y-Z√-1..

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(4)

From whence we conclude that a- -b√ −1 is a root of (2); that is, all the roots of (2) are obtained by changing in those of (1) the sign of √-1. The real roots, according to this, must be the same in the two equations.

We may now consider the following beautiful proposition as demonstrated from the foregoing.

PROPOSITION XI.

An algebraic equation which has real coefficients is always composed of as many real factors of the 1° degree as it has real roots, and of as many real factors of the 2° degree as it has pairs of imaginary roots.

DEPRESSION OR ELEVATION OF ROOTS OF EQUATIONS.

PROPOSITION.

251. To transform an equation into another whose roots shall be the roots of the proposed equation increased or diminished by any given quantity.

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Let ax+A¬1+Аçx12+ · • An−1¤+A2=0, be an equation, and let it be required to transform it into an equation whose roots shall be the roots of this equation diminished by r.

This transformation might be effected by substituting y+r for x in the proposed equation, and the resulting equation in y would be that required; but this operation is generally very tedious, and we must therefore have recourse to some more simple mode of forming the transformed equation. If we write yr for r in the proposed equation, it will obviously be an equation of the very same dimensions, and its form will evidently be

ay"+B1y"-1+B2y"-2+.....B2-13+B1=0

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(1)*

in which B1, B2, &c., will be polynomials involving r. But y=x-r, and therefore (1) becomes

a(x-r)"+B1(x-7)"1..... B2-1(x−r)+B=0.. (2) which, when developed, must be identical with the proposed equation; for, since y+r was substituted for x in the proposed, and then r-r for y in (2), the transformed equation, we must necessarily have reverted to the original equation; hence we have

a(x-r)+B,(x-r)"1+.. B-1(x—r)+B1=ar"+A,"1+.. A2-12+A ̧•

*

It will be of the same form with the development in the note to (Art. 239). We give it again below, arranged according to the powers of r instead of y. After substituting y+r for x, we write the development of each term of the proposed equation in a horizontal line; the first horizontal line is the development of ax", the second of An-1, and so on.

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In which the first column is of the same form as the proposed equation; the second column, or coefficient of r, is derived from the first by multiplying the coefficient of each term by its exponent, and diminishing the exponent by unity; the third column, or coeffi

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cient of is derived from the second in a similar manner, and so on.

1.2'

If we designate by f(x) the first member of the given equation, and by f'(x) the first derived function, by f'(x) the second derived, and so on, we shall have

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Now, if we divide the first member by r-r, every term will evidently be divisible, except the last, B., which will be the remainder, and the quotient will be a(x-r)+B,(x-r)"+..... B1-2(x-7)+B-1;

and since the second member is identical with the first, the very same quotient and remainder would arise by dividing this second member also by r-r; hence it appears that if the first member of the original equation be divided by r―r, the remainder will be the last or absolute term of the sought transformed equation.

Again, if we divide the quotient thus obtained, viz.,

a(x-r)+B1(x-r)2+.... B1-2(x—7)+B2-1

by r-r, the remainder will obviously be B-1, the coefficient of the term last but one in the transformed equation; and thus, by successive divisions of the polynomial in the first member of the proposed equation by x-r, we shall obtain the whole of the coefficients of the required equation.

RULE.

Let the polynomial in the first member of the proposed equation be a function of x, and r the quantity by which the roots of the equation are to be diminished or increased; then divide the proposed polynomial by x-r, or 1+7, according as the roots of the proposed equation are to be diminished or increased, and the quotient thus obtained by the same divisor, giving a second quotient, which divide by the same divisor, and so on till the division terminates; then will the coefficients of the transformed equation, beginning with the highest power of the unknown quantity, be the coefficient of the highest power of the unknown quantity in the proposed equation, and the several remainders arising from the successive divisions taken in a reverse order, the first remainder being the last or absolute term in the required transformed equation.

Note. When there is an absent term in the equation, its place must be supplied with a cipher.

EXAMPLES.

(1) Transform the equation 5.x-12x3+3x2+4x-5=0 into another whose roots shall be less than those of the proposed equation by 2. x-2) 5x-12x+3x2+4x-5 (5x3-2x2-x+2

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This laborious operation can be avoided by Horner's Synthetic Method of division, and its great superiority over the usual method will be at once apparent by comparing the subsequent elegant process with the work above. Taking the same example, and writing the modified or changed term of the divisor x-2 on the right hand instead of the left, the whole of the work will be thus arranged:

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..5y+28y+51y+32y-1-0 is the required equation, as before.

(2) Transform the equation 5y'+28y3+51y2+32y-1=0 into another having its roots greater by 2 than those of the proposed equation. 5+28+ 51 +32-1 (−2

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...5x-12x3-3x2+4x-5=0 is the sought equation, which, from the transformations we have made, must be the original equation in Example 1. (3) Find the equation whose roots are less by 1.7 than those of the equation 23-2x2+3x-4=0. 1-2 3-4 (1

1-1 2

2

0

2

1

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