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5. If 2s = a+b+c+d, what is the sum, and what the product of s a, - b, sc, s d? and what is the sum of their squares?

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6. Let several binomial factors, of which a is the first term, and where a, b, c, d, ... are the second terms of the several factors, be multiplied together: then describe the manner in which the coefficients of the several powers of x in the product are formed of the quantities a, b, c, d, ...

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7. Prove that a2 = (a b + c) (a + = (a+b+c) (—a+b+c); and hence that 4a2b2 (−a+b+c) (a−b+c) (a+b—c).

y5 by x

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8. Divide 5+ y5 by x + y, and x5 y; and show that if any other odd whole number be substituted for 5 in these expressions, the division will terminate without a remainder.

9. Convert (u + x + y + z)2 into the form (u + x)2 + (u + y)2 + (u + z)2 + (x + y)2 + (x + 2)2 + (y + z)2 · 2 (u2 + x2 + y2 + z2) ;

and likewise into the form

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{{(u+x+y)2 + (u+x+z)2 + (u+y+z)2 + (x+y+z)2 — (u2+x2+y2+z2)} ; and again into

u2 + (2u+x) x+{2 (u+x) + y} y + { 2 (u+x+y) + z}z:

and show that in this last, u may also change its place with either of the other quantities, x, y, or z.

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10. Multiply a + b √ 1 by b√ 1, and also by c + d√ 1; and multiply together four factors, each equal to a+b√✓ — 1, and then four others, each of which is a-b√√—1; and lastly, multiply the factors a+b√−1,c+d√−1, e+f√−1, y+h -1, and i+k-1 together.

11. If the term rectangle of two lines, in the first ten propositions of the second book of Euclid, be exchanged for the term product of two numbers, and square on a line for the square of a number; show that the propositions thus transformed are also true, whatever those numbers may be.

12. Divide a by 1 +✓ 3, and by (− 1 + √

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3)2 and show that the quotient in the latter case is the same as would be obtained if we divide a by

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3)3a (−1 + √ − 3)3.. 13. Show that the sum of x (x + y + z), y (y + z + x), and z (z + x + y)

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and a2+(-b)2 + (−c)2 + 2a (b) + 2a (−c) + 2 (—b) (c).

THE GREATEST COMMON MEASURE AND LEAST COMMON MULTIPLE OF TWO OR MORE POLYNOMIALS.

A common measure of two or more quantities, whether expressed algebraically or arithmetically, is any quantity which will divide them both without a re'mainder.

The greatest common measure is the greatest quantity which will divide them without a remainder.

A common multiple of any number of quantities is any quantity which is divisible by them without remainders.

The least common multiple is the least quantity that is divisible by them all without remainders.

Quantities are said to be prime to each other which have no common measure, except unity.

I. To find the greatest common measure of two quantities.

RULES.

1. If there be any visible factor of one of the terms, whether it be the numerator or denominator, which is not a factor of the other, it may be rejected as forming no part of the common measure.

This applies more especially to monomial factors, as it is not often easy, except in very simple cases, to detect binomial or higher factors.

On the same principle, if it will facilitate the future operation, any factor may be brought into either of the terms.

2. Range both expressions in ascending or descending (no matter which, but descending is most usual) powers of some one quantity concerned in the expressions. Divide the greater by the less and the less by the remainder; then the remainder by the previous one, and so on till the work terminates by giving no remainder. The last divisor is the greatest common measure of the fraction. 3. If more than two expressions be given, of which to find the greatest common measure, proceed as directed in the corresponding subject in arithmetic, pp. 42-4.

PROOFS AND REMARKS.

....

1. DENOTE them by X and X,, and let the following series of operations be performed, where Q, Q1, Q2 Q are the successive quotients, and X2, X3, Xm Xm+2 the corresponding successive remainders. The first column indicates the operation according to the arithmetical type, and the second expresses continually that divisor × quotient + remainder = dividend.

....

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Now suppose X+2 to be the remainder which becomes 0: then X, = Qm Xm+1, and Xm+, is the last remainder, which, therefore, divides X, exactly. Substitute this value of X, in the preceding equation:

then Xm-1 = Qm-1 Qm Xm+1 + Xm+1 = Xm+i {Qm−1 Qm + 1}

From which, as Q-1 and Q, are integral, we see that X-, is divisible by Xm+1 exactly.

m-1

Next substitute the values of X-, and X, in the third equation from the end, in terms of Xm+ 19 and we have

Xm-2 = Qm-2 {Qm−1 Qm + 1} Xm+1 + Qm Xm+1

Qm−2

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Whence as Qm, Qm_1, and Q-2 are integral, X-2 is divisible by Xm+1.

......

m-19

Whence Xm+1

....

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In the same way, we find that in succession all the preceding Xm-3, X m-49 X4, X3, X2, X1, X, are divisible by Xm+1° is a common measure of the given expressions X and X1; and it is likewise a common measure of all the subordinate quantities, or remainders, X2, X ̧ · 2. In the next place, Xm+1 is the greatest common measure. be any other common measure, and put X : = mY and X, =nY. Then from the foregoing equations, and substituting these values of X and X,,

X2 = X — QX, =

X, Q,

{m – nQ} Y

= P1Y,

X1 = X1 — Q1 X2 = {n — mQ, + nQQ,} Y = P2Y,
X1 = X2- Q2 X2 = {..

X3

......} Y = P,Y,

=

Xm+1 = Xm-1-Qm, Xm = {..........} Y PY,

For, let Y

where it is evident from the composition of the coefficients P1, P2, .... Pm, that they are all integers, since they are composed of the products, sums and differences of the integers m, n, Q, Q1, Qm, by hypothesis. Hence we have PY = X+, and P, integral; and therefore Y is less than X+1: that is, Xm+1 is the greatest common measure.

....

3. It also follows from this, that any common measure Y of two terms is also a measure of their greatest common measure. For since PY = X„+, and P„ is an integer, Xm+s is divisible by Y without remainder.

4. The quotients

X

and

Xatı

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may be thus formed by a succession of

operations; and each is a closer approximation than the preceding to the true

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that is, multiplying each successive value by the quantity under which it stands, and adding the second preceding one to the quotient.

These expressions have several curious and interesting properties, which there is not room in this work to touch upon. One or two, however, will be stated under Continued Fractions, a little further on.

5. As a form of work we may adopt with advantage that given on the corresponding occasion in the arithmetic, p. 43; and, generally speaking, we may avail ourselves of the method of detached coefficients. Also, to avoid the introduction of fractional coefficients in the successive divisions, we may, by cross-multiplication of the coefficients of the highest terms of divisor and dividend, reduce the leading coefficients to identity.

6. When the given expressions can visibly be resolved into factors, it is always better to do so to the utmost possible extent. The factors which are common to both are common measures; and if the resolution has been complete, all the common measures will thus have been obtained; and their continued product will be the greatest common measure.

EXAMPLES.

Ex. 1. Find the greatest common measure of the expressions æ

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= (x2 — xy + y2) (x + y) (x + y) (x − y) and

=

X1 = x2 + x3 y2 — x2 y3 — y5 — (x3 — y3) (x2 + y2) x5

= (x − y) (x2 + xy + y2);

and we see that the only factor common to both terms is a —

therefore the greatest common measure.

Ex. 2. To find the greatest common measure of a3 a2 + 2ab+b2) a3. ab2 (a

a3 + 2a2b + ab2

-

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ab2 and a2 + 2ab + b2.

- 2a2b — 2ab2) a2 + 2ab + b2 (

or dividing by — 2ab, which is not a divisor of the other quantity,

a+b) a2+2ab + b2 (a + b

a2 + ab

ab + b2

ab + b2

Therefore a + b is the greatest common divisor.

But by detached coefficients, and under the indicated arrangement, (art. 5,) it would stand thus:

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Hence la + 1b, or a + b is the greatest common measure.

Ex. 3. Find the greatest common measure of æo + 3æ3. 6x4 6x3+9x2+

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And as there is no remainder, we have by restoring the letters a3 for the common measure of X and X1.

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Ex. 4. Find the greatest common measure of a2 4 and ab+2b.
Ex. 5. And of a5 a3b and a1 .b1.

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-

Ex. 6. And of a3x + 2a2x2 + 2ax3 + x1 and 5a5 + 10a1x + 5a3x2.
Ex. 7. And of 6x5 + 20x1.

12x3

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48x2+22x + 12 and x6 + 4005 3x4

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1. If the quantities be prime to each other, their least common multiple is their product.

2. If one of them be a multiple of all or any of the others, whatever is a multiple of this is a multiple of those others; and the least common multiple which takes in this greater quantity will be the least common multiple of all those others.

....

3. Let X, XI, XI be the quantities, no one of which is a multiple of any of the others; and let the greatest common measure of X and X, be V, such that X = mV and X1 = nV. Then m and n are prime to each other; and the least common multiple M of X and X, is mnV, for it is the least quantity divisible by mV and nV, or by X and X1. But mV. nV XX, V

M = mnV=

=

product of the quantities. V their greatest com. meas.

Again, the least common multiple of X, X1, and X, is the least common multiple of M and X. Suppose V1 to be the greatest common measure of M and X, then the least common multiple of X, X1, and X1, is

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Proceeding in the same manner, we find for p quantities

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product of their g. c. measures.

EXAMPLES.

Ex. 1. Required the least common multiple of a3 + a2b and a2 — b2.
Here a+b= V, is their greatest common measure.

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Ex. 2. Required the least common multiple of x3 + x2 + x + 1 and 3

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Here the greatest common measure is x2 + 1 = V, and hence

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Ex. 3. Required the least common multiple of a2+ab, a1+a2b2 and a2—b2.

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Hence again V1, the greatest common measure of M and X1, or of a2 (a + b) (a2 + b2) and a2 b2, is a+b; and therefore

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