where p, q, r, s are any quantities positive or negative which do not contain x.

To prove this, it is in the first place clear that any common factor of A and B is also a factor of pA +qB and of rA + sB.

So also, any common factor of pA+qB and rA+sB is also a factor of s (pA+qB) − q (rA+sB), that is, of (sp-qr) A. Hence, as (sp-qr) does not contain x, any common factor of pA+qB and rA+sB must be a factor of A, provided only that p, q, r, s are not so related that spqr0. Similarly any common factor of pA + qB and rA + SB is also a factor of r (pA + qB) − p (rA + sB), that is of (rq-ps) B, and therefore of B.

Therefore the H.C.F. of A and B is the same as the H.C.F. of pA+ qB and rA + sB.

Ex. To find the H. C. F. of 2x1 + x3 – 6x2 - 2x + 3 and 2x1 — 3x3 + 2x −3.
We have, by subtraction,

The required H.C.F. is the H.C.F. of (I) and (II), and therefore of (I) and

Multiply (III) by 2 and add (I), and we have another expression,

namely

4x3- 2x2-6x=2x (2x2 - x − 3)

(IV),

such that the H.C.F. of (III) and (IV) is the H.C.F. required. But the H.C.F. of (III) and (IV) is obviously 2x2 - x - 3.

99. If R, S...... be the successive remainders in the process of finding the H.C.F. of the two expressions A and B by the method of Art. 97; then, as we have seen, every common factor of A and B is a factor of R, and therefore a common factor of A and R. Similarly every common factor of A and R is a common factor of R and S. And so on; so that every common factor of A and B is a factor of every remainder, and therefore must be a factor of the H.C.F. Hence every common multinomial factor of two expres

S. A.

6

sions is a factor of their highest common multinomial factor; and this is obviously true also of monomial factors. Therefore every common factor of two expressions is a factor of their H.C.F.

100. The H. C. F. of three or more multinomial expressions can be found as follows.

Let the expressions be A, B, C, D,....

Find G the H.C. F. of A and B.

Then, since the required H. C. F. will be a common factor of A and B, it will be a factor of G: we have therefore to find the H.C.F. of G, C, D....

Hence we first find the H. C. F. of two of the given expressions, and then find the H.C.F. of this result and of the third expression; and so on.

101. The highest common factor of algebraical expressions is sometimes, but very inappropriately, called their greatest common measure (G.C.M.).

If one expression is of higher dimensions than another, in a particular letter, we have no reason to suppose that it is numerically greater: for example, a is not necessarily greater than a; in fact, if a is positive and less than unity, a2 is less than a.

It should also be noticed that if we give particular numerical values to the letters involved in any two expressions and in their H. C. F., the numerical value of the H.C.F. is by no means necessarily the G.C.M. of the values of the expressions. This is not the case even when the given expressions are integral for the particular values chosen. For example, the H.C.F. of 14x2+15x+1 and 22x2 + 23x+1 will be found to be x+1; but if we suppose to be, the numerical values of the expressions will be 12 and 18, which have 6 for G.C.M., whereas the numerical value of the H. C.F. will be 3.

5.

x2 - 4y2 + 12yz-9z2 and x2 + 2xz − 4y2 + 8yz — 3z3.

6. 20a - 3a3b+b and 64a* - 3ab3 + 5ba.

7. a3-a2b+ab2 + 14b3 and 4a3 + 3a2b - 9ab2 + 2b3.

8. 2x2+x3-9x2+8x-2 and 2x-7x3 + 11x2 - 8x + 2. 11x-9ax3- a2x2 - a1 and 13x - 10ax3 – 2α2x2 — a1. x2+x3- 9x2 - 3x + 18 and x5 + 6x2-49x + 42.

9.

10.

11. x1- 2x3 + 5x2 - 4x + 3 and 2x1 — x3 + 6x2 + 2x + 3. 12. x2+3x2 + 6x + 35 and x2 + 2x3- 5x2 + 26x + 21.

LOWEST COMMON MULTIPLE.

102. Definitions. A Common Multiple of two or more integral expressions, is an expression which is exactly divisible by each of them.

The Lowest Common Multiple of two or more integral expressions, is the expression of lowest dimensions which is exactly divisible by each of them.

Instead of Lowest Common Multiple it is usual to write L.C.M.

103. When the factors of expressions are known, their L.C.M. can be at once written down.

Consider, for example, the expressions

a3b2 (x − a)2 (x — b.)3 and ab1 (x — a)a (x — b).

It is clear that any common multiple must contain a3 as a factor; it must also contain b*, (x - a)* and (x-b). Any common multiple must therefore have a b (x - α)1 (x — b)3 as a factor; and the common multiple which has no unnecessary factors, that is to say the lowest common multiple, must therefore be a3b1 (x − a)* (x — b)3.

From the above example it will be seen that the L.C.M. of two or more expressions which are expressed as the product of factors of the first degree, is obtained by taking every different factor which occurs in the expressions to the highest power which it has in any one of them.

Ex. 1. Find the L. C. M. of 3x2yz, 27x3y2z2 and 6xy224.

Ans. 54x3y2z4.

Ex. 2. Find the L. C. M. of 6ab2 (a+b)2 and 4a2b (a2 – b2).

Ex. 3.

Ans. 12a2b2 (a + b)2 (a - b). Find the L. C. M. of 2axy (x -- y)2, 3ax2 (x2 - y2) and 4y2 (x+y)2. Ans. 12ax2y2 (x2 — y2)2.

Ex. 4. Find the L. c. M. of x2 - 3x+2, x2 - 5x+6 and x2 - 4x + 3.

Ans. (x-1) (x-2) (x − 3).

104. When the factors of the expressions whose L.C.M. is required cannot be seen by inspection, their H.C.F. must be found by the method of Art. 97.

and

Thus, to find the L. C. M. of x3 + x2 - 2 and x3 + 2x2 - 3.

The H. C. F. will be found to be x-1;

x3+x2 - 2 = (x -- 1) (x2+2x+2),

x3+2x2-3=(x − 1) (x2+3x+3).

Then, since x2+2x+2 and x2+3x+3 have no common factor, the required L. C. M. is (x − 1) (x2+2x+2) (x2 + 3x+3).

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105. Let A and B stand for any two integral expressions, and let H stand for their H.C.F., and L for their

L.C.M.

Let a and b be the quotients when A and B respectively are divided by H; so that

AH.a and B= H.b.

Since H is the highest common factor of A and B, a and b can have no common factors.

and B must be H × a × b. Thus

L=H.a.b.

Hence the L.C.M. of A

also

B....

.(i);

L × H = Ha × Hb = A × B.................................. (ii).

From (i) we see that the L.C.M. of any two expressions is found by dividing one of the expressions by their H.C.F., and multiplying the quotient by the other expression.

From (ii) we see that the product of any two expressions is equal to the product of their H.C.F. and L.C.M.

2. 4a2-5ab + b2 and 3a3 — 3a2b + ab2 — b3.

3.

3x13x2+23x - 21 and 6x3 + x2 - 44x + 21.

4. x11x2 + 49 and 7x1 - 40x3 + 75x2 - 40x+7.

5.

x+6x+11x+6 and x2+x3- 4x3- 4x.

6. x2x2 + 8x - 8 and x2 + 4x3 - 8x2 + 24x.

7. 8a3 - 18ab2, 8a3 + 8a3b – 6ab2 and 4a2 – 8ab + 3b2.