A

TREATISE

O F

ALGEBRA.

PART II.

Of the Genefis and Refolution of Equations of all Degrees; and of the different Affections of the Roots.

CHA P. I.

OF THE GENESIS AND RESOLUTION OF EQUATIONS IN GENERAL; AND THE NUMBER OF ROOTS AN EQUATION OF ANY DEGREE MAY HAVE.

$ 1.

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FTER the fame manner as the higher

powers are produced by the multiplication of the lower powers of the fame root; equations of fuperior orders are generated by the multiplication of equations of inferior orders involving the fame unknown quantity. And an K

equation

equation of any dimenfion may be confidered as produced by the multiplication of as many simple equations as it has dimenfions; or of any other equations whatsoever, if the fum of their dimenfions is equal to the dimenfion of that equation. Thus any cubic equation may be conceived as generated by the multiplication of three fimple equations, or of one quadratic and one fimple equation. A biquadratic as generated by the multiplication of four fimple equations, or of two quadratic equations; or laftly, of one cubic and one fimple equation.

§ 2. If the equations which you fuppofe multiplied by one another are the fame, then the equation generated will be nothing else but fome power of those equations, and the operation is merely involution; of which we have treated already and, when any fuch equation is given, the fimple equation by whofe multiplication it is produced is found by evolution, or the extraction of a root.

But when the equations that are fuppofed to be multiplied by each other are different, then other equations than powers are generated; which to refolve into the fimple equations whence they are generated, is a different operation from involution, and is what is called, the refolution of equations.

But as evolution is performed by obferving and tracing back the fteps of involution; fo to

discover

discover the rules for the refolution of equations, we must carefully obferve their genera

tion.

3. Suppofe the unknown quantity to be x, and its values in any fimple equations to be a, b, c, d, &c. then thofe fimple equations, by bringing all the terms to one fide, become x-a=0,x− b = 0, x — c = o, &c. And, the product of any two of these, as x — a × xbo will give a quadratic equation, or an equation of two dimenfions. The 'product of any three of them, as xa x x -b x x = c o will give a cubic equation, or one of three dimenfions. The product of any four of them will give a biquadratic equation, or one of four dimenfions, as x -a xx = o. And, in general, "In the equation produced, the highest dimension of the unknown quantity will be equal to the number of fimple equations that are multiplied by each other."

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b x x -CXX

d

$ 4. When any equation equivalent to this biquadratic x-axx-bxx-cxx_d = o is proposed to be refolved, the whole difficulty confifts in finding the fimple equations x-a=0,

x—b=0,x—co,x-d=o, by whofe multiplication it is produced; for each of these fimple equations gives one of the values of x, and one folution of the proposed equation. For,

if

any

6 xx

of the values of x deduced from thofe fimple equations be fubftituted in the propofed equation, in place of x, then all the terms of that equation will vanifh, and the whole be found equal to nothing. Becaufe when it is supposed that x = a, or x = b, or x = c, or x = d, then the produc x a x x -d does vanish, because one of the factors is equal to nothing. There are therefore four fuppofi tions that give xaxx — bxx — cxx — d = o according to the propofed equation; that is, there are four roots of the propofed equation. And after the fame manner, "Any other equation admits of as many folutions as there are fimple equations multiplied by one another that produce it," or "as many as there are units in the highest dimenfion of the unknown quantity in the propofed equation."

§ 5. But as there are no other quantities whatfoever befides these four (a, b, c, d,) that substituted in the productx-axx-bxx-cxx-d, in the place of x, will make the product vanish; ther fore, the equation xa x xbxx-c xx-d=o, cannot poffibly have more than these four roots, and cannot admit of more folutions than four. If you fubflitute in that product a quantity neither equal to a, nor b, nor c, nor d, which fuppofe e, then fiace neither ea, e-b, e—c, nored is equal to nothing;

their product ea xe - b xe - cxe. cannot be equal to nothing, but muft be fome real product; and therefore there is no supposition befide one of the forefaid four, that gives a juft value of x according to the proposed equa

tion. So that it can have no more than thefe four roots. And after the fame manner it appears, that "No equation can have more roots than it contains dimenfions of the unknown quantity."

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$ 6. To make all this ftill plainer by an example, in numbers; fuppofe the equation to be refolved to be x4-10x3+ 35x250x +24=0, and that you difcover that this equation is the farne with the product of x-xx-2x-3 xx4, then you certainly infer that the four values of x are 1, 2, 3, 4; feeing any of these numbers placed for x makes that product, and confequently **

• 50x + 24,

equal to nothing, according to the propofed equation. And it is certain that there can be no other values of x befides thefe four: fince when you fubftitute any other number for x in thofe factors x1, x2, x — 3, x — 4, none of the factors vanish, and therefore their product cannot be equal to nothing according to the equation.

§ 7. It may be useful fometimes to confider equations as generated from others of an infeK 3

rior