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an-2

an-1 = sum of the roots, with their signs changed;

=sum of their products taken two at a time, with

their signs changed;
&c. = &c.
a=product of all the roots, with their signs changed;

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1
If an-m+1=am-1, then a, being a root, is also a root.

a, If the coefficients are alternately positive and negative, the roots are all positive; if the coefficients are all positive, the roots are all negative.

If the signs of all the terms are changed, the roots are not changed.

If the signs of the alternate terms beginning with the second are changed, the signs of all the roots are changed.

If the coefficients are integers, the equation cannot have a fractional root.

If any coefficient be altered, all the roots are altered.

If n is odd, Q(x)=0 has at least one possible root, which has the same sign as — a.

If n is even, and a negative, the equation has two possible roots, one positive, the other negative.

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If a is positive, there is an even, and if negative, an odd, number of positive possible roots.

If a + v ß is a root of an equation of which the coefficients are rational, ß not being a complete square, then a-VB is also a root.

(W. 280—97; Bour. 275-80; L. 178--94.)

TRANSFORMATION OF EQUATIONS.

(33.) To transform the equation P(x)=0 into another whose roots are a, -C, 0, -C, &c. Qn-c: assume x=y+c, and by substituting this value of w in P(x)=0, we obtain

n(n-1) y" + nc y"-1+

c

y" – + &c. + (n − 1)an-10

1

2

+ An-1

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which may for convenience be thus expressed : Yon -(0)

Yon -? (c) y" +

yn-1 + 1.2...(n-1)

( ¥*(c) y(c)

y+(c)=0; 1.2

or $(y+c)=0.

1.2... (n— 2)%*-+&c.

y +

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To take away the mth term of (x)=0: assume the coefficient of the mth term of the above equation,

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hence, to take away the mth term we must solve an equation of m - 1 dimensions.

If m=2,

nc + an-1

= 0;

m=3,

n (n − 1)

c? + (n − 1) an-10+ An-2 = 0. 1. 2

To transform an equation into one whose roots are respectively m times the roots of the original equation: multiply the successive terms by 1, m, m*, &c.

If the coefficients be respectively divided by 1, m, m, &c. the roots will all be divided by m.

To transform the equation q (x) = 0 into one whose roots

1 are the reciprocals of the roots of the former : assume x=

Y we obtain

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(W. 280—97; Bour. 275-80; G. 310—25.)

ß To take away the coefficient of yn -m+, assume

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ELIMINATION.

(34.) If two equations each containing two unknown quantities, have a common divisor containing both unknown quantities, the number of solutions is infinite.

If they have a common divisor containing only one unknown quantity as X, X will have a finite, y an infinite number of values.

To eliminate one of the unknown quantities as x: arrange both equations according to the powers of X, and proceed as in finding the greatest common divisor; we shall at length obtain a remainder containing only y, which being made =0, will give the final equation required.

If this equation be capable of solution, the values of x may be found by substituting those of y obtained from it, in the preceding remainder.

If we have three equations (1), (2), (3), containing three unknown quantities, x may be eliminated between (1) and (2), and (1) and (3); and two equations thus obtained, each containing y and %, from which y may be eliminated; the result will be the final equation in terms of x only.

The same method may be applied to n equations, containing n unknown quantities.

(Bour. 281-90; L. 184—96.) If b, b,, &c. are the factors that have been introduced for the purpose of avoiding fractional quotients, the last remainder divided by b .bg ... &c. will be the true final equation.

In general, the degree of the final equation equals the number of separate systems of values that satisfy the given equations.

If the last remainder is independent of the unknown quantities, the equations are incompatible.

Further observations, and their application to particular

(Bour. 361–84.)

cases.

Another method. Let the equations be reduced to the form

1 + $7.80 + AqX? + AzQ3 + + Am 20.=0, (1)

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(:+++...+2) = 0, + + + &c. (a.

) *= *a, +

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303

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a, +

+

+

+ &c.

-1

xn.

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