150. The methods of resolving expressions into factors, given in the last chapter, often enable us to shorten the work of finding the H. C. F. required.

1. Find the H. C. F. of

x2 + 3 x3 + 12 x − 16; x3 — 13 x + 12.

Both of these expressions vanish when 1 is put for x. both are divisible by x

-1, § 135.

Therefore,

The first quotient is x3 + 4x2 + 4x + 16 = (x2 + 4) (x + 4).
The second quotient is x2 + x

12 = (x 3) (x+4).

Therefore, the H.C.F. is (x − 1) (x + 4).

2. Find the H. C. F. of

2x2+9x3 +14x + 3; 3x2 + 14 x3 + 9 x + 2.

2x+9x3 + 14x + 3|3 x

The remainder, x3 — 24 x
The quotient of x3 - 24 x

2

+ 14x3 + 9x+2

6x + 28 x3 + 18x +43

6x4 +27 x3 + 42x+9

- 5, vanishes when 5 is put for x.

5 divided by x-5 is x2+5x+1. Since 5 is not an exact divisor of 3, x 5 is not a factor of 2x4 +9x3 + 14 x + 3; but x2 + 5x + 1 is found by trial to be a factor, and is, therefore, the H. C. F. required.

3. Find the H. C. F. of

28 x2+39x+5; 84 x8-16 x 60 x 8.

- -

By § 132, the factors of 28 x2 + 39 x + 5 are 7 x + 1 and 4x + 5. The factor 7x + 1 is the H. C. F. required.

4. Find the H. C. F. of

2x46x3x2+15x-10; 4x+6x-4x2-15x-15. 2x46x3x2 + 15 x 10 4x4 + 6 x3- 4x2 - 15x-15|2

4 24 12x32x2 + 30 x 20

18 x3- 2 x2 - 45 x + 5

The remainder = 2 x2 (9x-1) — 5 (9 x — - 1)

=(2x25) (9x-1).

The factor 2 x2-5 is the H. C. F. required.

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

— .

xy2

6 x 11 x2y + 2 y3 and 9 x3-22 xy2-8 y3.

6 x3- 11 x2y

6 ე3

8 x2y - 4 xy2

3x2y + 4xy2+ 2 y3

3x2y + 4 xy2 + 2 y3

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

-

(6 x3 — 11 x2y + 2 y3) ÷ (3x2 - 4 xy - 2 y2) = 2xy. (9 x3 — 22 xy2 — 8 y3) ÷ (3 x2 – 4 xy − 2 y2) = 3 x + 4 y.

.. the L. C. M. (2 x − y) (3 x + 4 y) (3 x2 — 4 xy — 2 y2).

=

EXERCISE 50.

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

1. 4x2+3x-10; 4x3 +7x2 - 3 x 15.

2. 2x3-6 x2+5x-2; 8x3- 23 x2 + 17 x 6.

3. 6 x3-7 ax2 - 20 a2x; 3x2+ax - 4a2.

4. 3 x3-13x2+23x-21; 6x8 + x2 - 44 x + 21. 5. c2c+c; 2c2c82c-2.

/6. a3 — 6a2x + 12 ax2 – 8 x3; 2 a2 − 8 ax + 8 x2.

7. 7 x3- 2x2-5; 7x8+12x2+10x + 5. 8. x13x2+36; xx3- 7 x2+x+6. 9. 2x+3x2-7x-10; 4x3- 4x2 - 9 x + 5. 10. 12x8x230 x16; 6x32x2 - 13 x 6. ~11. 6x3 + x2-5x-2; 6 x3 + 5 x2 - 3 x - 2.

12. x9x2+26x-24; x3- 12x2 + 47 x - 60.

13. 4x3- 2x2-16 x 91; 12x8 - 28 x2 - 37 x 42.

14. x44x8 +10 x2-12x+9; x++ 2x2+9.

15. 2x3-3x2-16x +24; 4x+2x-28x3-16x2-32x.

16. 12x+4x2 + 17x-3; 24x3-52x2 + 14x-1. 17. 2 x3 +7 ax2 + 4 a2x − 3 a3; 4x3 +9 ax2 — 2 a2x — a3. 18. 2x3-9ax2+9a2x-7a8; 4x3-20 ax2+20 a2x-16 a3. 19. 2x+9x3 +14x+3; 3x2+14x3 +9x+2.

20. 20x+2x2-18x + 48; 20 x 17 x2 + 48 x 3. 21. 2x+x2 - 12 x + 9; 2 x3-7 x2 + 12 x

22. x8x+3; x3-3x2 + 21 x

-

8.

9.

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

24. 8x-6x - x2+15x-25; 4x3+7x2-3x-15. 25. 4x3- 4x2-5x+3; 10 x2 - 19 x + 6.

26. 6x13x + 3x2 + 2x; 6x-10x+4x2-6x+4. 27. 2x-3x2 + 2x2 - 2x-3; 4x4 + 3x2+4x-3. 28. 3x4x3- 2x2+2x-8; 6x+13x2+3x+20. 29. 3x+2x+x2; 3x2 + 2 x3-3x2+2x-1.

30. 3 2x+5x2+2x3; 12 - 17 x + 2x2 + 3x3.

х

31. 10x6x2 - 11x3 +9x4 - 6x5;

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

32. x4x14x2+x+1; x5 - 4x4 - x3-2x2+8x+2.

33. 2a2a-3a2-2a; 3 at a3 - 2 a2 - 16 a.

-

34. 6x-14ax2+6a2x-4a3; x2 - ax3- a2x2 - a3x - 2a1. 35. 4 2x 8x2+7x3-9x5; 2+5x-10x2 - 7x2+6x1.

36. 2a+3a3x-9 a2x2; 6ax 3 ax1- 17 a3x2+14a2x2.

37. 2a54 at + 8 a 12 a2 + 6a;

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. C. M.

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 ab F 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.

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 L and 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 L and 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. 6x2x2; 2x2+7x-4; 2x2 - 7x+3.

2. a2+2ab+b2; a2 — b2; a3 + 2 a2b + 2 ab2 + b3. 3. x2-5 ax+4a2; x2 - 3 ax+2a2; 3x2-10 ax + 7 a2. 4. x2 + x − 6; x3 − 2 x2 − x + 2 ; x3 + 3 x2 − 6 x − 8.

5. x86 x2 + 11x-6; x3- 8x2+ 19 x 12;

x8-9x2+26 x 24.

6. 6x2+7xy-3y2; 3x2+11xy - 4y2;

2x2+11 xy + 12 y2.

7. 8-14 a6a2; 4 a + 4a2 - 3 a3; 4a2+2 a3 — 6 a*.

8. 6x8 +7x2 - 3x; 3x2+14x-5; 6x2 +39 x + 45. 9. 27 x3- a3; 6 x2+ax - a2; 15x2-5 ax + 3 bx — ab.

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

х

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

- — 12. 12x2+2x-4; 12 x242x24; 12 x2 28 x 24.

13. 2x2+3x-5; 3x2-x-2; 2x2+x-3.

2

х

14. x2+7x2+5x-1; x2+3x-3x3-1;

3x2 + 5 x2 + x − 1.