expression, in reference to the letter to which that exponent belongs.

In like manner, the expressions x2y-3 xy+5xy2-2, x2y2+2xy2—3x+1, &c. are of the third and fourth dimensions, or degree, respectively, as concerns x and y, the dimensions being marked by the highest sum which the exponents of these letters furnish in any term of which they are factors.

Of two divisors of the same dimensions, that is said to be the greater which is divisible by the greatest numerical factor. The rule for finding the greatest common divisor of two polynomials is as follows:

:

Divide one polynomial by the other; the lower by the higher, when they differ in dimensions: if there be no remainder, the polynomial, taken for divisor, is the greatest common measure of both.

But if, as usually happens, there be a remainder, take it for a new divisor, and the former divisor for a new dividend, and proceed with these either till the operation terminates, or till stopped by a second remainder.

As before, take this remainder for a divisor, and the immediately preceding divisor for a dividend, and proceed in this way so long as remainders successively arise; the last divisor, or that which gives no remainder, will be the greatest common measure of the two polynomials.

The truth of this rule depends chiefly upon these two principles:

1. If a quantity divide another, it will divide any multiple of that other. 2. If a quantity divide each of two others, it will divide their sum and their difference.

The first of these principles is self-evident; if a is a divisor of b, it must, of course, be a divisor of any number of times b. The second principle is proved thus:-Let c be a divisor of both a and b, and put m and n for the respective quotients: then, since the dividend is always equal to the product of quotient and divisor, we have amc, and b=nc; so that a+b= mcnc, and a-b-mc-nc; and each of these is divisible by c, the quotient in the one case being m+n, and in the other case m―n.

Let a represent the polynomial taken for dividend, and b that taken for divisor, and imagine the work in the margin to be carried on according to the directions in the rule, c being put for the first remainder, d for the second, and so on, till the remainder at length becomes 0, and the operation terminates. That the last divisor is the greatest common measure of a and b may be thus proved.

Since, by the first principle, whatever divides a and b must divide a and br, and, by the second, whatever divides these must divide their difference, c, it follows that, whatever divides a and b, divides c.

In like manner, whatever divides b and c divides b and cs; and, whatever divides these, divides also their difference, d; therefore, whatever divides a and b, also divides c and d. Similarly, whatever divides c and d, divides c and dt, as also their difference, e: hence, whatever divides a and b, divides also c, d, and e.

b]a [r

br

es

d]c[t dt

e]d[u

eu

0

Thus every common divisor of a and b, is also a common divisor of the series of remainders c, d, &c. Let e be the last of these, then e divides d:

therefore it divides dt; therefore it divides edt or c; therefore it divides cs; and e being thus a divisor of d and cs, it is a divisor of d+cs or b; e, therefore, divides both c and b; hence it divides c+br or a. It follows, therefore, that the last remainder, e, divides a and b; and as it was before proved that whatever divides a and b must also divide e, it further follows that e is the greatest common divisor of a and b; for no quantity greater than e can divide e. And thus the above process necessarily leads to the greatest common measure of the proposed quantities, a and b.

NOTE.-In carrying on the operation here described, the following particulars should be attended to:

1. If any simple factor be common to all the terms of either of the divisors, but not common to those of the corresponding dividend, this factor may be expunged from the divisor, and the simpler result employed instead.

2. Or if the terms of a dividend have a common factor which does not also enter every term of the corresponding divisor, this factor may, in like manner, be expunged.

3. And we may always introduce into either dividend or divisor, any factor which does not enter the other.

That we may make these changes without error, will appear from considering that the common measure of the first dividend and divisor, a, b, is also the greatest common measure of each dividend and divisor: and the factors common to any two quantities can never be interfered with by either multiplying or dividing one of them by a quantity, no factor of which enters the other. We have spoken above of expunging simple factors, only because we could but seldom discover, by mere inspection, whether a compound factor of one of the expressions enters the other expression or not.

From the observation (3) above, it is plain that we may always avoid fractional terms in the quotients and remainders, since we may always multiply the dividends by such quantities as will preclude their entrance.

We shall now give an example or two of the application of the above rule, and shall then show how a little attention to the principles previously taught would enable us frequently to dispense with it, even in cases not remarkable for their simplicity.

(1) Find the greatest common measure of 7x2-12x+5 and x2—6x+5. x2-6x+5]7x-12x+5 [7

Or, rejecting the factor 30,

7x2-42x+35

30x-30

x-1]x2-6x+5[x—5
x2-x

5245

Hence, the greatest common measure is x-1.

(2) Find the greatest common measure of

12x3-4x2-3x-1 and 8x3-4x2—2x+1.

Multiplying the first expression by 2, to prevent the entrance of a fraction in the quotient, we have

Hence, the greatest common measure is 4x3—1. (3) Find the greatest common measure of

9x+53x2-9x-18 and x2+11x+30.

x2+11x+30]9.x3-53x2 -9x-18[9x—46

9x399x2+270x

-462-279x -18

Therefore, the greatest common measure is x+6.

Therefore, the greatest common measure of numerator and denominator is a2-x2; so that, dividing both by this, the fraction, in its lowest terms,

45. To show how these and similar examples may be otherwise treated, let us return to example (1). In this we observe that one of the expressions is of the second degree: the two factors of the first degree, which produce it, and which we may denote by x-a, x-b, we know to be such that a+b=6 and ab=5 (see page 44): hence, these factors will be found if we can discover two numbers, a and b, such that their sum is 6 and their product 5: it is at once obvious that the numbers are 5 and 1. Hence, the factors of x2-6x+5 are x-5 and x-1; so that, if a common measure exists, it must be one or other of these factors: upon trial it is found to be x-1.

In example (3), also, one of the expressions is of the second degree, namely, x2+11x+30: the two numbers whose sum makes 11 and product

30, are soon seen to be 5 and 6: hence, the factors are x+5 and x+6: the former cannot be a factor of 9x+53x2-9x-18, because 5 is not a factor of the last term 18: hence, if a common measure exists, it must be x+6: which upon trial is found to divide the expression of the third degree.

In example (4), the numerator is (a2+x2) (a2-x2), and the denominator a(a2x2)+x(u2—x2)=(a+x) (a2x2): hence, a2—x2 is the greatest factor common to both.

The expression of the third degree, in example (3), might have been decomposed in a way analogous to this: thus,

9x3+53x2—9x—18—9x2(x+6)—(x2+9x+18)=9¿2(x+6)—(x+3)

(x+6);

from which we see that the expression is divisible by x+6. And example (2) may be treated in a similar way.

It must be observed that the mode of decomposing an expression of the second degree here employed, and which is suggested by what is noticed at page 44, requires that the coefficient of x2 be unit. If the coefficient be any other number we must divide the whole by that number, and then decompose the quotient as above: thus, if 7x-12x+5 be proposed, we 5 And it is easily seen that and 1 are

12

may write it 7 X ·x+

[This trial method of decomposing an expression of the second degree into its simple factors, will be hereafter replaced by a general rule for that purpose. The subject belongs to the theory of quadratic equations.]

46. The greatest common measure of three quantities is obtained by first finding, as above, the greatest common measure of two, and then the greatest common measure of this and the third quantity. When four quantities are proposed, we may find the G.C.M. (greatest common measure) of two; then the G.C.M. of the remaining two, and, lastly, the G.C.M. of the results thus found: and so on for any number of quantities.

47. To find the least common multiple of two or more quantities:

The least common multiple of a set of quantities is the least quantity, that is, the quantity of lowest degree, which is divisible by each of them. If the quantities are numbers, it is the least number divisible by each. The initial letters L.C.M. stand for least commor multiple. Let a and b represent any two quantities, and m their G.C.M.: then a and b are both divisible by m. Let the quotient, in the former case, be p, and, in the latter, q: then a pm, and b=qm: and p and 9 cannot have a common factor. Hence, their product, pq, will be the least quantity which contains them both; so that the least quantity which contains pm and qm, will be pqm: that is to say, pqm is the least quantity which contains a and b, or their least common multiple; so that, to find the L.C.M. of a and b, we have only to divide their product by their G.C.M.

And the L.C.M. of three quantities may be found by taking the_L.C.M. of two, and then the L.C.M. of this and the third, and so on.

EXAMPLES.

Required the G.C.M. of the following expressions:—

(1) x3+2x2+2x+1 and x2+2x+1.
(2) 2x3+x2—8x+5 and 7x2—12x+5.
(3) 4x3—2x2—3x+1 and 3x2-2x-1.

(4) 6x3—6x2+2x-2 and 12x2—15x+3.

(5) 2r*—11x2+12 and 3x3-48x.

(6) x*—4x3+8x2—16x+16 and xa—6x3+13x2—12x+4.

Required the L.C.M. of the following expressions:

(1) x3+1 and x2+2x+1.

(2) r3—y3 and x2—y2.

(3) 2x3+x2—2x−1 and 2ƒ3—x2—2x+1.

(4) x—1, x2—1, x—2 and x2—4.

(5) 3(x2—ax), 4(x2+ax) and 5(x2—aa).

The learner has now been conducted through a course of operations sufficiently extensive to justify the expectation, that all the fundamental rules of algebra have become familiar to him, and that he has acquired some expertness in the management of algebraical expressions. We shall, therefore, proceed to show the application and use of the principles and processes with which he has been occupied in actual calculation, where definite arithmetical meanings will be given to symbols which have been employed, in what has preceded, without any regard to special interpretation.

CHAPTER III

ON SIMPLE EQUATIONS.

48. An Equation is merely a statement, in algebraic characters, that two quantities are equal, the one to the other. Thus, 7+3=10, is an equation: it states that the quantity, 7+3, is equal to 10. In like manner, if 3x-5 be equal to 2x+3, then, introducing the sign of equality between the two, we have an equation, namely, 3x-5=2x+3. By the solution of an equation is meant the interpretation of the unknown quantity or quantities, involved in the equation, proper to justify the statement of equality: thus, in the equation, 3x-5=2x+3, the