4. (15 a2bc-27 ab2c33 abc2)+(-3 abc).

5. (17 ab 13 ab2) ÷ 2 ab.

+

15. D= 15 a2 - 9a5 + 18 ao, d = 3 a2, find q.

16. (8 ab 24 ab3 + 16 a7b8)÷(-8 a1b).

19. (36 x3y2-24 x2y2z - 18 xy2z) ÷ (— 6 xy2).

20. (36 a10-24 ao + 21 a3)+(- 6 a3).

21. (100 a2bc-75 ab2c+ 50 abc2)+(-25 abc).

22. (35 c2xy + 42 cx2 - 56 cxy) ÷ 7 cx.

24. (12 an+3-15 an+2-27 an+1)+(-3a").

25. (12 an+3-15 an+2-27 an+1)+(-3 an+1).

an+1)÷(−3

26. Show that (5n+3 + 5n+2 + 5n+1)÷ 5" = 155.

27. (a2x+3b=+3 - 2 a2+16x+1)÷(— a2*b*).

28. (22n+322n+2)÷22n+1

31. [(a+b)x+(a + b)y]÷(a+b).

32. [r2(m + n) — 2 r(m + n) + (m + n)]÷(m + n). 33. [12 x2(a + b)3 — 32 xy(a + b)2]÷[− 4 x(a + b)2].

34. [2 m2(x — y2)3 — 3 m(x − y2)2 — (x − y2)] ÷ (x − y2).

35. [−8 a2b(x − y)2 + 9 ab2(x − y)]÷ ab(x − y). 36. [x3(α2 + b2) — 2 x2(a2 + b2)]÷x2(a2 + b2).

37. [12 b(x2 — y2) — 15 b2(x2 — y2)] ÷ 3 b(x2 — y2).

DIVISION OF A POLYNOMIAL BY A POLYNOMIAL

157. This kind of division will be understood best by studying an example.

Divide 237 x2+10x-8 by x-2.

2 x3 7 x2+10x 8x

1.

-3x2+10x 8

3x(x-2)=-3x2+ 6 x

5.

6. 4(x-2)=

8

8

2

1. Both dividend and divisor are arranged in descending powers of x. 2. The first term of the dividend is divided by the first term of the divisor to obtain the first term of the quotient, 2x2. The entire divisor is then multiplied by the first term of the quotient.

3. The product obtained is subtracted from the dividend.

4. The first term of the remainder is divided by the first term of the divisor, to obtain the next term of the quotient, - 3 x. The entire divisor is then multiplied by this second term of the quotient.

5. The product is subtracted from the last remainder.

6. The process described in the last two steps is repeated until, in exact division, a remainder zero is obtained.

158. The explanation just given may be regarded as a rule for the division of a polynomial by a polynomial. It is of the greatest importance that a proper arrangement of the terms of the polynomials be made at the beginning and that the same arrangement be observed in all the remainders obtained in the course of the work.

Let the student explain how the next term of the quotient is obtained. Also explain all the operations involved in steps 5 and 6.

To check examples in long division the relation d· q = D may be used, or arbitrary values of the letters may be substituted. If the latter method is employed, values of the letters should be chosen which will not make the divisor 0.

7. (3x2-423 +20)+(x-2).

8. (6 a3-23 a2b+ 25 ab2 - 6 b3)+(2a-3b).

9. (30 ap-6 bp +12 cp)÷(5 a-b+2c).

10. (20 ac- 15 ad - 12bc9bd)÷(5 a − 3b).

11. (3 abd - 3 cd + abc- c2)+(ab — c).

12. (6 a3b +9 ab2 + 3 abc + 2 a2c + 3 bc + c2)÷(3 ab+c).

13.

(24+23-4x2+5x-3)÷(1 − x + x2).

14. (27 28 y3)+(3x-2y).

15. (8 a3b3 — c3ď3) ÷ (4 a2b2 + 2 abcd + c2d2).

16. (a2 + b2 + c2 + 2 ab - 2 ac - 2bc)÷(a + b — c).

17. (5 a 15 a3+5a+15)+(a + 3).

18. (2a-6 a3+3a2-3 a + 1)÷(a2 - 3 a + 1).

19. (42 a+ 41 a3 - 9 a2 - 9 a 1)÷(7 a2 + 8a+ 1). 20. (2 m1-6 m3 +3 m2 - 3 m +1)+(m2 - 3m + 1). 21. (6 a3x 17 a2x2+14 ax3-3x1)+(2a-3x).

Ꮖ

22. (2 x1 + x3y − 13 x2y2 — 3 xу3 + y1) ÷ (x2 − 2 xy — y2). 23. (15 a 10 ab+4a3b2+6a2b3 - 3 ab1)÷(5a3 + 3 ab2). 24. (21 a1 - 16 a3b + 16 a2b2 — 5 ab3 + 2 b1) ÷ (3 a2 — ab+b2).

25. (20 a 53 a7+45 a9-a8)÷(4 a2 5 a3).

—

26. (x55x1y-10 x2y3 +10x3y2+ 5 xy1—y3)+(x2-2xy + y2). 27. (a1 +2 a2x2 + x1 — b1) ÷ (a2 + x2 + b2).

c2)

28. (6a2 + ab +7 ac-12 b2+ 19 bc-5c2)+(2a+3b-c). 29. (15 x2-29 xy + 12 y2 — 22 yz — 60 z2) ÷ (5 x − 3y+10 z). 30. (48 x2y-80x3y3 — 8 xу5 + 200 x1y2)÷(20 x2y2 — 4 xy3). 31. (343 a3x3- 64 b3x6)÷(49 a2x2+28 abx2 + 16 b2x1).

+16

32. (20x1+32 x 51 x3- 12x2)÷(4 x2 -7x-8). 33. (32 a2+45 b2 + 60 c2 + 76 ab + 88 ac+ 104 bc)

(8a9b+10 c).

34. (1.2 a2+1.17 ab - 11.34 b2)÷(1.5 a + 5.4 b). 35. [x2+(a+c)x + ac]÷(a + x).

36. [y2 (a - b)y — ab]÷ (a − y).

160. Division with a Remainder. If the dividend is not the product of the divisor multiplied by some integral algebraic expression, we shall have a remainder.

1. Divide 6213x3 by 2x+1.

Unless otherwise directed, perform all divisions in descending powers of some letter, and continue the division until the exponent of the highest power of the letter of arrangement in a remainder is less than that of the highest power of that letter in the divisor.

EXERCISE

161. Divide, and check by the relation d· q+ r =D.

1. (x2 - 3x+5) ÷ (x + 1).

2. (4-3x2+2x)+(2+x).
3 x2 + 2 x)÷(2 + x).

3. (2-1)+(x2 — x+1).
−1)÷(x2.

4. (3 aa3+2)+(1 - a2).

÷

5. (7 a2+6 a3+5a-7)÷(3a-1).

6. (7 a2+6 a3+5a-7)÷(2 a2 + 3 a + 2).

7. (23-8a3-2 a2x)÷(2 a- x).