5. Find the H. C. F. and the L. C. M. of:

6 x3 - 11 x2y + 2 y3 and 9 x3 — 22 ху2 - 8 y3.

... the H. C. F. = 3 x2 - 4 ху - 2 у2.

To find the L. C. M., divide each of the expressions by the

H. C. F.

(6 x3 - 11 x2y + 2 y3) ÷ (3x2 - 4 ху — 2 y2) = 2 x - y. (9 х3 - 22 ху2 – 8 y3) ÷ (3x2 - 4 ху — 2 y2) = 3 x + 4 y.

... the L. C. M. = (2x - y) (3 x + 4 y) (3 x2 - 4 ху — 2 у2).

EXERCISE 50.

Find the H. C. F. and the L. C. M. of:

1.4x2 + 3 x − 10; 4x3 + 7 x2 - 3 x - 15.
2.2 x3 - 6 x2+5x-2;8 x2 - 23x2 + 17 х
3.6 x3 7 ax2 - 20 a2x; 3x2 + ax – 4 а2.
4. 3 x2 - 13 x2 + 23 x - 21 ; 6 x3 + x2 - 44 х + 21.

6.

5. с-2 c3 + c; 2c2 - 2 c3 - 2 с 2.

6. d3 - 6a2x + 12 ax2 - 8 x3; 2a2-8ax +8 x2.

7.7 x3 - 2x2 - 5; 7 x2 + 12x2 + 10 x + 5.

8. x2 - 13 x2 + 36; x2 - x3 - 7 x2 + x + 6. 9.2 x3 + 3x2 - 7 x - 10 ; 4x3 - 4x2 - 9 x + 5. 10. 12 x3 x2 - 30 х - 16; 6 x2 - 2 х2 - 13 х - 6. 11.6 x3 + x2 - 5x - 2; 6 x3 + 5 x2 3 x 2. 12. x3 - 9 x2 + 26 x - 24; x3 12 x2 + 47 x

42.

13. 4 x - 2x2 - 16 х — 91 ; 12 x3 — 28 х2 - 37 х 14. x2 - 4x2 + 10 x2 - 12 x + 9; x2 + 2x2 + 9. 15. 2x3-3x2 - 16x + 24; 4x5 + 2x2 - 28 х3 - 16 x2 - 32 х. 16. 12 x2 + 4 x2 + 17 x - 3; 24 x3 - 52x2 + 14 х – 1. 17.2 x3 +7 ax2 + 4 a2x - 3a2; 4x3 +9 ах2 - 2 а2х — а3. 18. 2x2 - 9ax2+9a2x - 7a3; 4x3- 20 ax2 + 20 a2x – 16 a3. 19. 2x2 + 9 x2 + 14x + 3; 3x2 + 14x3 + 9x + 2. 20. 20 x2 + 2x2 - 18 x + 48; 20 x2 - 17 x2 + 48 х – 3. 21. 2 x3 + x2 - 12 x + 9; 2 x3 - 7 x2 + 12 х — 9.

22. x3- 8x + 3 ; x - 3x5 + 21 x - 8.

23. 3x3 - 3x2y + xy2 - y3; 4x3 — х2у - 3 ху2.

24. 8 x2 - 6 x3 - x2 + 15 x - 25; 4x3 + 7 x2 - 3x — 15.

25. 4 x8 - 4x2 - 5x +3; 10 x2 - 19 x + 6.

26. 6x4 - 13 x2 + 3x2 + 2x ; 6x2 - 10 x2 + 4x2 - 6x + 4. 27. 2x2 - 3x2 + 2x2-2x-3; 4x2 + 3x2 + 4 x - 3. 6x8 +13 x2 + 3 x + 20.

28. 3x2 - x - 2x2 + 2x -8; 29.3x5 + 2x2+x2; 3x2 + 2 x

− 3 x2 + 2 x - 1.

30.3-2x+5x2+2x3; 12 - 17 x + 2x2 + 3 x3.

31. 10x - 6 x2 - 11 x3 + 9 x2 - 6x5;

60 x + 4x2 + 10 x3 + 10 x2 + 4 x5.

32. x2 - x3 - 14x2 + x + 1; x5 - 4x4 - x2 - 2x2 + 8x + 2. 33. 2α - 2a3-3a2-2a; 3a - a3- 2 а2 — 16 а. 34. 6x8 - 14ax2 + 6a2x - 4a3; x2 - ах3 - а2x2 - а3x - 2a2. 35.4-2x- 8x2 + 7x3 - 9x5; 2+5x- 10x2 - 7x8 + 6x4. 36. 2a2+3a3x - 9a2x2; 6 ах - Зах2 - 17 a3x2 + 14a2x3. 37. 2a5 - 4a2 + 8 a3 - 12 a2 +6a;

2

3 - 3a5 - 6 a+ + 9a3 – 3 а2. 151. The product of the H.C.F. and the L. C. M. of two expressions is equal to the product of the given expressions.

Let A and B stand for any two expressions; and let F stand for their H. C. F. and M for their L. С. М.

Let a and b be the quotients when A and B respectively are divided by F.

Then

Since F stands for the H.C.F. of A and B, F contains all the common factors of A and B. Therefore, a and b have no common factor, and abF is the L. C. M. of A and B.

Put M for its equal, abF, in equation (1), and we have

The lowest common multiple of two expressions may be found by dividing their product by their highest common factor, or by dividing either of them by their highest common factor and multiplying the quotient by the other.

153. The H. C. F. of three or more expressions is obtained by finding the H. C. F. of two of them; then the H. C. F. of this result and of the third expression; and so on.

For, if A, B, and C stand for three expressions,
and D for the highest common factor of A and B,
and E for the highest common factor of D and C,
then D contains every factor common to A and B,
and E contains every factor common to D and C;
that is, every factor common to A, B, and C.

A

154. The L. C. M. of three or more expressions is obtained by finding the L. C. M. of two of them; then the L. C. M. of this result and of the third expression; and so on.

For, if A, B, and C stand for three expressions,

and L for the lowest common multiple of A and B,
and M for the lowest common multiple of Land C,
then L is the expression of lowest degree that is

exactly divisible by A and B,

and M is the expression of lowest degree that is exactly divisible by Land C. That is, M is the expression of lowest degree that is exactly divisible by A, B, and C.

EXERCISE 51.

Find the H. C. F. and the L. C. M. of:

1.6 x2 + x -2; 2x2 + 7x - 4; 2x2 - 7x + 3.

2. a2 + 2ab + b2; a2-b2; a3 + 2a2b+2 ab2 + b2.

3. x2 - 5 ах +4 a2; x2 - 3 ax + 2a2; 3x2 - 10 ах + 7a2.
4. x2 + x - 6 ; x - 2x2 - x + 2; x2 + 3x2 - 6x - 8.
5. x3 - 6 x2 + 11 x - 6; x3 — 8 x2 + 19 x — 12;

x3- 9 x2 + 26 x - 24.

6.6 x2 + 7 ху - 3y2; 3x2 + 11 ху — 4 у2;

9. 27 x3 - α3; 6 x2 + ax - а2; 15 х2 — 5 ax + 3 bx - ab.

10. x2 - 1; 2x2 - x - 1; 3x2 - x - 2.

11. 6 x2 - x - 2 ; 21 x2 - 17 x + 2; 14x2 + 5 x - 1.

12. 12 x2 + 2 х - 4; 12x2 - 42 х - 24; 12 x2 - 28 х - 24.

13.2x2+3x-5; 3x2 -x-2; 2x2 + x - 3.

14. x2 + 7x2 + 5x - 1; x2 + 3x - 3x8 1;

CHAPTER IX.

FRACTIONS.

Definitions.

155. An algebraic fraction is the indicated quotient of

two expressions written in the form

a

156. The dividend a is called the numerator; the divisor b is called the denominator; the numerator and denominator are called the terms of the fraction.

157. If the numerator and denominator of a fraction are

both multiplied by the same number, or both divided by the same number, the value of the fraction is not altered.