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EXERCISES.

I.

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1. What operation bears the same relation to multiplication that multiplication bears to addition ?

2. Explain what is meant by a reverse operation in Algebra, and give an example.

3. Explain the meaning of the terms factor and dimension as applied to Algebraical expressions. Give examples.

4. What is meant by the rule of signs in the multiplication of Algebraical quantities ? State whether you regard the rule as conventional or demonstrative, giving your reason in the former case, or a demonstration in the latter. 5. Define the meaning of am; show that a".a” = a*+, and

*-*; and explain the notations a", a ".

6. Explain and interpret the expressions ab, and abc, when a, b, c, represent straight lines.

7. Find the products of 5abx and 4cdy; mab and -Nc; – 12abc? and 5ab'd?; -12abc and -8abd respectively.

8. Divide 20abcdxy by 4cdy; -mnabc by mab; -60a%b8%d by - 12a2bc); +96a*b*%d by -8abod®, respectively.

% 9. Multiply 9a+ 2x by 3a ; 4a-86 by --3ab; 2-axta by - ar; 3a– 4ab-56% by -6a26*c; max-nay+pas by axyz ; -aʻx+b*y—c*3! by x*yoz?, respectively.

10. Divide 27a' + bax by 3a; -12ab+24ab' by - 3ab; -ax+a'r"-aixby-ax; 18a*b*c* +24a*b*c* +30a2b4eby-6a-boc"; ma*x'ys-na-xy's +pa rys by axyz; ~a*r*y*z*+b.c*y*zo—c*x*yoz by x*yoz?, respectively.

11. Shew the truth of, and express in words at length, the identical expressions : 1. a(a+b)+(a+b).

2. ala-6)—b(a6). 3. (a+b)'+(a−b)* = 2(a+b). 4. (a'+b)-(ab)' = 4ab. 12. In the first ten propositions of the second book of Euclid's Elements, if “the rectangle contained by two lines" be exchanged for “the product of two numbers,” and “the square on a line" for the square of a number :” shew that the propositions so changed are true for all numbers whatever.

13. In the multiplication and division of Algebraical polynomials, what preliminary steps are necessary in order to avoid confusion in performing these operations ?

II. Find the following products, and prove the correctness of them by reversing the operation in each example :

1. Of 2x+3y and 2x+3y; 2x-3y and 2x + 3y; 2x-3y and 2x-3y.

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2. Of ? +2x+7 and x+5; 3.2°-5.xy+7y and 2x-3y.
3. Of a +ar+r" and as-ax+x?.
4. Of 8a-12ab+18ab-2788 and 2a+36.
5. Of a?- 2ab +26+ and a' + 2ab +26*.
6. Of 5x -7.0-3.++8 and 72° +8.
7. Of x' +62° +24.0+60 and 2-66° +12.1-60.
8. Of x-3ax' +2a*x and x—2ax—2a”.
9. Of 81x* +27x*y+9x'yo+3xy + y and 3x-y.
10. Of x + x3 + x +x+1 and x-*+1.
11. Of af-2a+3a-2a+1 and a +2a+3a +2a+1.
12. Of 1-5x+10.co -10.0° +5.4* - * and 1-2x+x.
13. Of x-3-3-9 +4x-1-8 and 2-3+x-? +4x-'+8.
14. Of x-3-3-'y-'+x-'y-?-y- and x-'+y-?
15. Of — 3x-5 + 2x-*y- by —2x-5—3x-y-.
16. Of 5x*y-5_7xy-8 +4x-'y-1—3x-*y+x-*y8 by xoyxy.

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III. Determine the following quotients, and verify their correctness :1. +10x +35.xo+50x+24 by X+4. 2. x+x - 4x° +53-3 by x + 2x-3. 3. 256.c* +16.x"y"+y' by 16x + 4xy+ya. 4. a'--3265 by a-26. 5. —x*y+xPyxy+xy*—ys by —yo. 6. 21.2-2.c* - 70.x*—23. +33.c+27 by 7x +4x-9. 7. 50.x*yo-67 +25xy'-45x'y: - 41c*y+20.x' by 5xy' -3y - 4x*y+5.24. 8. 21x*y* +25x*y$ +68xy* —40y8—56.30 — 18x*y by 5y2—8.x2—6xy. 9. 40ab" +49a%b2—65ab3—250*6+6a'—2565 by 5ab2 +3a' -8a2b-568. 10. +3.r+9 by x+2. 11. 20–2x-5.0 +20.0°-25x2° +14x-3 by x + 2x-3. 12. x-x*y: -2°* ++ by x-roy-xy+y!. 13. x+r-'+1 by :+2-1+1. 14. a'r-9-6-471-26-1-6by ax-'+6-2x+6. 15. 2xy-3-5x*y-? +7x*y-l-5x2 + 2xy by x*y-3—x*y-+xy-'.

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IV. Find the quotients arising from the division of :1. a-3ab+3ab2-63 by a-b. 2. **+(4abb)x-(a-2)(a +36) by x-a+26. 3. (x+y)+3xy(1-3-y)- 1 by x+y-1. 4. a+63 +03-3abc by a+b+c. 5. x-(a*—.—c)x?—(6c)ax+bc by x— ax+c.

6. (a-1)2-(a tas-2).«* +(4a+3a+2x-3(a+1) by (a—1)xé—(a-1).«+3.

7. (ax+by+(ay-bx+cox? +cʻy by x +yo. 8. a'd'-a'y? +bysb*z* by ax+by+ay+bu.

9. x+y—x + 2xoyo—22°—1 by x++y—–1. 10. 3(a' +64) — 8ab(n° +6°)+14aby 3a:— 2ab +6%.

11. (1262—29bc+15co).«* +(2368—316?c—9bc? +150°).x® +(1064—66?c)x by (36-50x+26.

12. a +88-ab(a + b)(ab-5a-6)} by a: -6.
13. (a - b)2-(x-ba +(2-a)6 by ax - bx+a-69.

14. (a! +3a*x+9ax? +27x*) (as - 3a*x+9a.co - 272°) by a- 18a%x* +81x*.

15. (23 – a°)6+(x2 - a®)ax+(x − a)a by (a+b)(x – a). 16. xy+xyz X’y wyʻz+x37

; – x78 +2492 2yz' by y+% – 4. 17. xy +2cyz*+ya+yz* - 2xyʻz – xz'— yozby xy - xz+yz. 18. (yz-a)-(ca y)(ab - -*) by (bc - 1)(y--a)-(3-by)(y-cz).

19. (ax+by)+(ax by)+(bu-ay)+(ay+bx)* by (a+b)*x* — 3ab(x2 - yo).

V. 1. From the expression (a+b)(ab)=a:—6, find the difference of the squares of 79 and 81, and of 1184 and 1211.

2. “To multiply two unequal numbers. Take the sum of the two numbers and multiply half of it into itself. From this product subtract the square of half the difference of the two numbers" (Kholasat-al-Hisab). Give the Algebraic formula which expresses this, rule, and state it in words at length in the simplest form.

3. Subtract the product (x-7)(x-1) from (x-2)*(x-3), and add to the result (x-1)×(1-2).

4. What value of x will make the difference between (x + 1)(x+2) and (1-1)(x-2) equal to 54 ?

5. Write the Algebraical expressions for the square of the sum of the cubes of a and b, and the cube of the sum of their squares, and find their product.

6. Determine by inspection the common factor in each of the terms of the expression (2+a%)e-2ax(2*—a) +ac+a).

7. Multiply 3+:01: by 1–x, and shew that the value of the product is .0009 when x =:1.

8. Divide 2.091.x* +9.22.2° +3.694.1-1.2 by 1.02.x+4.

9. If a, b, k be positive integers, ascertain whether (a+k):—(6+k) is divisible by a_b.

10. What number must be added to the expression 23+x*—4(x+3) that it may be divisible by x—6 without remainder ?

11. If the dividend be 4ab2+2(3a*—264)—ab(5a—1162) and the quotient be 2(a+b)a+(a?—?), what is the divisor?

12. Write at length the forms of the four products (mx+a)(nx+b), and exemplify the use of them. 13. Shew that (1) ax'+bx+cx+d is divisible by x +b, if ad = ac.

(2) axt+bx+c by z + , if ac = b.
(3) 2—59.0+4r by (r—c), if q' = r4.

14. Multiply a+x, 6+y, c+ together, and deduce (x+y) from the result.

VI.

1. Find the continued product of 3+1, 2+2, 3—X, 4—~.

2. Find what 25+4x3 — 3x2 +2 becomes when x is changed into 2+5, and when x is changed into x-1.

3. Multiply x+a, x+b, x+c, x+d together, and deduce from the product the coefficient of x in the product (x+2)(x+6)(x+10)(x+14).

4. Multiply x +(x−y)?yo by x-(x+y)'+ya.

5. Show that the difference between (2° +2.c+3)(3.xo + 2x+1). and (2–2x+3)(3.co -- 2x+1) is the same as the difference between (x +4x+1) and (x2-4x+1).

6. Shew that (3.co—4x+2)-(2x* +90+3) is divisible by x +x+1 without performing the operation of division.

7. By what Algebraical expression must xx+ys be multiplied so that the product may be 2+x+y + x®yo +x+y8+xy4 +yo ?

8. Multiply 23—9xy+23xy* — 1548 by x7y and divide the quotient. by x-8xy+yo.

9. Write the coefficients of x' and of 2 in the product of 3.r-5.28 +72-9x+11 and +6.04+12.-12.c -- 6.0+1.

10. The product of two factors is (2.c +3y)+(2y+3:)and one of them is 2x+54 +32; find the other.

VII. Form the products of the following sets of factors without performing the process of multiplication.

1. a+b+c and a+b-c; a-b+cand 6+c-a; a+b+c and a+c-6; a+b+e and b+c-a; atb-c and a to-b; atb-c and b+c-a. 2. a+b+c+d and a+b+c-d; a+b+d-c and a+d-b+c;

a+b-c-d and a-b7c-d; a-bto-d and a+b+c+d;

b+c+d-a and a+d-6-o; a+b-c-d and ba+c-d. 3. a' tab +62 and a'-ab+b?. 4. 1-ax, 1-bx, and 1-ox.

VIII. Resolve the following expressions into their simple and quadratic factors :

1. (a +2a)-(-20).
2. alb+c)'+ba+c)+ca-6).
3. ab(a+b)+bcb+c)+ca(a +c)+2abc.
4. a_6_0+d2—2(ad+bc).
5. 4(ab+cd)-(a +82----d-).
6. (1-5)x+10(2-5)x+24.
7. 4(ad-bc)-(a'+d-6:-).

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IX. Perform the following divisions of 1. (a+b)(a +c)-(d+b)(d+c) by a-d. 2. a'(c-b)+ba-c)+(b-a) by a-b. 3. (ax+by+cz)-(bx+cy+az)' by (a+b)x+(6+c)y+(c+a)z. 4. a'(b+c)-ba+c)+((a+b)+ abc by a-b+c. 5. a8 —88 +aobo(a*—54) by (a' - ab+82)(a'+ab+b?). 6. a+b+c-3abc by a+b+c. 7. (+c)+(c+a)*+(a+b)-3(b+c)(c+a)(a+b) by a+b+c. 8. a'+b+0+d-3abc+abd+acd+bed) by a+b+c+d. 9. a(b+c)+b(a–c)+c+(a−b)+abc by a +b+c. 10. (b-ca +(c-ay+(ab)c by (a-c)(6-c).

11. a'(bo—c*)+(c'—a*)+ca-b?) and (a-6)* +(-c)+(-a) by (a-b)(b-c)(c-a). 12. a”(b+c-a):+*(c+a-6)*+c(a+b-c) by (ab).—2(a+b)c+c*.

X. Verify the truth of the following equivalent expressions : 1. (a+b)cd+(c+d)ab = (a +d)bc+(6+c)ad. 2. (a? +da)bc +(6+co)ad =(ac+bd)(ab+cd). 3. (a’ +62)(c +d) =(ac+bd)? +(adbc)? = (acbd)? +(ad+bc)'. 4. (a - b)(6d) = (ac-bd)-(ad be) = (ac+bd)-(ad+bc).

5. (a* +62 +62 +d) (x2 + y') = (ax+by) + (bx - ay)? +(xc+dy) +(dx-cy).

6. (a'+)(+d?)(x++y) = {(ac-bd).c+(ad+bc)y}+{(ac-bd y-(ad+bc).x}".

7. (a +69+*+da)(v? +x++y+z) = (av+bx+cy+dz)'+ (ax-bo+cz-dy)+(ay-cu-bz+de+(az-dv+by-cx)".

8. a*(1-0)+b(c--a)+((ab) = (a-b)(b-c)(c-a). 9. a (a-c)+(c-a)+ca-6)=(a-6)(1-c)(c-a)(a+b+c). 10. a'(1-0) +5c-a) +ca-b) =(a-6)(6—c)(c-a)(a +b+*+ab+ac+bc).

11. e(6—c)+85(ca)+ca-6) =(ab)(6—c)(c—a){a'+(6+c)a: +(bo+be+c)a+(6+c)(6+c?)}. 12. (a+b)-(a'+63) = 3ab(a+b).

(a+b)-(a'+b)=5ab(a+b)(a + ab +82).

(a+)-(a'+87) = 7ab(a + b)(a + ab +64). 13. (a+b+c)-(a+b+c)= 3(a + b)(b+c)(c+a).

(a+b+c)" - (a +65 +c") = 5(a+b)(b + c)(c+a)(a +62 +6+ab+ac+bc).

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