9. 2a+4a2+2a +4, a1-4a2-a-2.

10. 3-7 x2+4x-4, 2 x3-3x2+2x-8.

11. 2x2-x-1, 2 x2x2+x-1. x - x2+

12. 1-2a-5a2+6 a3, 1+5a+2 a2-8 a3.

13. 12 x3-30 x2-16 x, 6 x − 2 x3- 13 x2 - 6 x.

14. 6x2-13x-5, 18 x3-51 x2 + 13 x + 5.

15. 2+2x+2x+1, x3-4x-4x-5.

16. 2x2 x3+8x2-7x+2, x1-4x+3.

17. x+2x2-x+1, x2 + x3 + 2x2+x+1.

18. 2x+x3+4 x − 3, 3 x1 + 2 x3 − 2 x2 + 3 x −2.

--

.

19. y2+y-6, y3 — 2 y2 −y+2, y3 + 3y2 — 6 y −8.

20. 24-10 +35 x2- 50 x + 24, x3-8 x2 + 17 x 10. x3

21. 10 ama2 — 7 amx 33 am, 14 abax3 +11 abx2

15 abx.

22. abx+abx2-9 abx-9 ab, ax-7 ax2 - 6 ax.
23. x-2xy + 2 xу3 — y1, x* — 2 x3y+2x2y2 - 2 xy3 + y*.

24. 3x2-5x-3, x2+2x2-x-2, x3- 2x2-2x+1. 23 25. 2x2+3x+1, 2x2+5x+2, 2 x3 +5 x2-4x-3. 26. x2+x-12, 22-10x+21, 23-6a2-19x+84.

x3

27. 6x4x3-x, 4x3- 6x2-4x+3, 2x3 + x2 + x − 1. 28. 2x+5x2+5x+6, 3x2+5x2-5x-6, x2+4x2 + 5+2.

29. x2-11x+30, x3-12x2+41 x −30, x* — 12 x3 +47 ∞2 - 72 x + 36.

LOWEST COMMON MULTIPLE

168. The Lowest Common Multiple of two or more algebraic expressions is the expression of lowest degree that is exactly divisible by each of them.

Thus, a2b2c2 is the lowest common multiple of ab2c, abc2, and a2bc; that of (a+b)2 and a2 - b2 is (a + b)2 (a — b).

It will be observed that the exponent of each factor in the L. C. M. is the highest exponent that factor has in any of the given expressions. When the expressions contain numerical factors, the L. C. M. of these factors can be found as in arithmetic, and written as a coefficient of the algebraic lowest common multiple.

169. By Factoring Method.

.

When the given expressions can be easily factored, their lowest common multiple is usually found by inspection. It is necessary merely to take the product of all the common factors, giving to each the highest exponent it has in any of the given expressions, and to multiply this product by the L. C. M. of the numerical factors when there are any.

1. Find the lowest common multiple of a3b2c, a2b3d1, and a1d3. Each letter must appear in the L. C. M. with the highest exponent it has in any of the given expressions.

.. a4b3cd5 is the lowest common multiple.

2. Find the L. C. M. of a2-b2, a2+2ab+b2, and a2-2 ab + b2.

Find the L. C. M. of the following sets of expressions:

18. x2-2xy + y2, x2 — y2, x2+y3.
19. 26, x2+x-2, x2-4x+3.
20. x2+y2-22-2xy, x2-y2 — z2-2yz.
21. 1-x+x2, 1 + x + x2, 1+x2+x1.
22. a (x2-1), ab (x2-ax), a3b (x3-1).
23. (a+c) - b2, (a + b)2 — c2, (b+c)2 — a2.
24. x1 +5 x2y2+ 9 y1, x1 — 2 x3y + x2y2 — 9 y1.
25. xyxy, 3 xy(xy)2, 4 yz(xy)3.
26. x2 + xy + y2, x2 — xy + y2, x* + x2y2 + y2.
27. a1, a2-1, a3-1, a* - 1, a3 - 1.
28. a2-b2, a3 — b3, aa — b1, a3 — b3,

-

ao — bo.

29. x, x — y3, x2 — y2, x2-2xy + y2, x2 + 2xy + y2.

170. By Means of Highest Common Factor.

The highest common factor of two expressions contains all their common factors; hence, if it is removed from the expressions, the quotients thus obtained will contain all the factors not common. Consequently the product of these quotients and the H. C. F. will be the lowest common multiple.

Thus, the factors common to a2bx and ab3y are a and b, and their H. C. F. is ab. Removing this factor, we obtain the quotients ax and b2y. Hence ab × ax × b2y, or a2b3xy, is the L. C. M.

Let x and y be any two algebraic expressions, f their highest common factor, a and b the quotients when x and y respectively are divided by f; so that

x = fa, y=fb.

Since f, the H. C. F. of x and y, contains all their common factors, a and b can have no common factors; hence their That is, the H. C. F. of two expressions multiplied by their other factors is their L. C. M.

L. C. M. must be fba.

But fba is equal to y multiplied by a; hence the L. C. M. of two expressions may be obtained by dividing either of them by their H. C. F. and multiplying the quotient by the other.

Thus, the H. C. F. of ax2y and bxy2 is xy; ax2y ÷xy = ax. ax × bxy2 = abx2y2, the L. C. M.

Then

1. Find the lowest common multiple of a3 — 6 x2 + 11 x − 6 and 3-9x2 + 26

x − 24.

:. (x3 − 9 x2 + 26 x ·

24) (x 1) is the lowest common multiple.

It will be observed that the H. C. F. is x2 - 5x+6, and that x - 1 is the quotient obtained by dividing one expression by the H. C. F.

The lowest common multiple 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 a third expression, and so on.

Find the L. C. M. of the following sets of expressions:

14. 4a3-3a2-8a-1, a*-— a3-4a2-a +1.

15. 6x2-x-1, 2x2+3x-2.

16. aa3+8a-8, a3+4a2-8 a +24.

19. x2+7x+9, x2-3x+7, x2-2x+11.
20. n2-1, n3 + n2 + n + 1, n3 — n2 + n − 1.
21. 4-10 +9, 24+10 x2+20 x2-10 x - 21.

22. x3 — y3, x3y — y1, y2(x − y)2, x2 + xy + y2.

23. 2a-a-2 a2-2a-1, 3a-4a+6a2-7a-8. 24. 2+6x+9x+4, x2+2-11 -12,

25. 2x2+2x-1, 3 x3- 4 x + 1, 2 x3 – 3 x + 1.

-7x-6.

26. 2-11x+3x2+10x, 3x-14 a-6x2+5x. 27. 3-13x+12, 5x2-17 +6, 5a2+3x-2.

28. xaxa2x2-a3x-2a, 3a3-7 ax2+3 ax-2 a3. — — 29. 23-5x+9x-9, x3-x2-9x+9, x-4x2-12 x 9. 30. +152+66x+80, 2x2+15x-8, 2 ax2+3 ax-2 a. 31. a3-6a2+11a-6, a3 - 9 a2 + 26 a 24, a 8 a2 + 19a-12.

32. Show that the product of the L. C. M. and the H. C. F. of two expressions is equal to the product of the given expressions.