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=7(a+b)(b+c)(c+a){a2+b2+c2+ab+ac+bc)2+(a+b+c)abc}.

XI.

Reduce to their simplest forms—

1. (a+b)(b+c)−(c+d)(d+a)−(a+c)(b−d).

2. a(a+b+c-d)+b(a+b−c+d)+c(a−b+c+d)+d(−a+b+c+d). 3. (a+b)2+(a+c)3+(b+c)2+(a+d)2+(b+d)2+(c+d)2.

4. (a+b+c+d)2+(a−b-c+d)2+(a−b+c−d)2+(a+b−c−d)2. 5. {(a-b-c)a-(a+c—b)b+(b+c)o+(a−c)c}(a—b).

6. (a2—bc)(b−c)+(b2—ac)(c−a)+(c2—ab)(a—b).

7. (a−2b)(a+2b)+(2b−3c)(2b+3c)+(3c−d) (3c+d).

8. (a+b+c)(a+b+d)+(a+c+d)(b+c+d)−(a+b+c+d)3. 9. (ab+cd) (c2+d2) + cd(a2 + b2 — c2 — ď2).

10. {(ac+bd)2+(ad—be)3}.{(ac+bd)3 — (ad+be)3}.

11. 4{(a3 — b3)cd+(c3 — d2)ab}2 + { (a3 — b3) (c3 — d3) — 4abcd}3.
12. (a3—b2+c2—d2)2+2{(ab−cd)2+(be+ad)2 — (ac — bd j2}.
13. (a+b)3+(b+c)3 +(c+d)3+(d+a)3+(a+c)3 + (b+d)3.
14. (a—b)3+(a+b)3+3{(a−b)2(a+b)+(a+b)3(a−b)}.
15. a(b+c)2+b(a+c)3+c(a+b)3—c2(a+b)—b2(a+c)—a2(b+c).
16. (a+b+c)3—(a+b—c)3—(a+c—b)3—(b+c—a)3.

17. (a+b+c+d)* − (a*+b*+c+d3)

− (a+b+d)* − (a+c+d)* − (a+b+c)* − (b+c+d)*

+(a+b)*+(b+c)*+(c+d)3+(a+d)*+(b+d)'+(a+c)*. 18. (b22+ad) (bc)(a−d) + (c2a2+b3d2) (c— a) (b−d) +(a2b3+c3d3)(a-b)(c-d).

-

19. (a+b+c+d)3 − (b+c+d)3—(a+c+d)" — (a+b+d) − (a+b+c)+ (b+c)+(a+o)+(a+b)+(a+d)3+(b+d)3+(c+d)3− (a3+b+c+d3).

XII.

Verify the following expressions:

1. 30ab (9a-8b)(5a+26)-(4b-3a) (15a+4b) = 4ab.

2. (ab+ac+bc)3— (a2b2+a2c2+b2c2)=2(a+b+c)abc.

3. a(b+c)(b2+c2−a3)+b(c+a)(c2+a2—b2)+c(a+b)(a2 +b2— c2) =2abc(a+b+c).

4. (a+b+c)'−(a+b)* − (b+c)* − (c+a)*+a*+b1+co =12abc(a+b+c).

5. a3(b+c-a)2+b3 (c + a−b)3 + c3 (a+b−c)2+abc(a2+b2+c3)+ (a2+b2+c2—bc— ca−ab)(b+c− a)(c+a−b)(a+b−c) = 2abc(bc+ca+ab). 6. (a+b-2c)+(b+c—2a)3+(c+a—26)3

=3(a+b-2c)(b+c−2a)(a+c−26).

7. 2(a+b+c)3—(a+b)3 — (b+c)3—(c+a)3+3abc =3(ab+be+ac)(a+b+c).

8. 8(a+b+c)3—(a+b)3—(b+c)3—(c+a)3

=3(2a+b+c)(a+2b+c)(a+b+2c).

9. (a+b+c)3+a3+b3+c3=(a+b)3 +(b+c)3+(c+a)3+6abc. 10. (a-b)+(b−c)3 + (c—a)3 = 3(a—b)(b—c)(c-a).

11. {(a—b)2+(b−c)3 +(c—a)3 } 3 = 2{(a−b)*+(b−c)*+(c—a)*}. 12. {(a—b)2+(b−c)3+(c—a)3}3—54(a—b)2(b—c)2(c—a)3

=2(a+b−2c)2(b+c—2a)2(c+a—26)3.

13. (a—b)*+(b−c)*+(c—a)* = 2{(a—b)2(c—a)2+(a—b)2(b—c)2 +(b−c) (c—a)*} = 2(a2+b2+c2—ab-bc-ac)3.

14. (b2+c2—a3+ab+be+ca)2 (c2—b2)+(c2+a2—b2+ab+be+ca)2(a3—c3) +(a2+b2—c2+ab+bc+ca) = (a2—b3)(b2—c2)(c23—a2).

15. a2(b+c)3+b2(c+a)2+c2(a+b)2+abc(a+b+c) +(a2+b2+c2)(bc+ca+ab)=(a+b+c)(b+c)(c+a)(a+b).

16. {2bc(a—b)—(b3+c2—a2)(a—c)}2+(a−c)2(b+c—a)(c+a—b) (a+b—c)(a+b+c) = 4abc{abc—(b+c−a)(c+a—b)(a+b−c)}. 17. a'(b+c)'+b (a+c)* +c' (a+b)* = 2b3c2 (a+b)2 (a+c)2 +2c3a2(b+c)2(b+a)3+2a2b2(a+c)2(c+b)2—16a2b2c2(bc+ac+ab). · 18. (a+b)2(b+c)2(c+a)2+2a2b3c2—a1(b+c)3—b1(c+a)3—c‘(a+b)3 =2(ab+be+ca).

XIII.

Show the identity in value of the following sets of expressions:1. (a—b)3+(c—a)(c—b), (b−c)2+(a−b)(a—c) and

(c—a)2+(b—a)(b—c).

2. (a+b)2+(b+c)2+(c+a)3—(a2+b2+c2), 3(a2+b2+c3){(a—b)2+(b−c)2+(c-a)"}, and {a-b)+(b−c)2+(c—a)2+6(ab+ac+bc)}.

3. ac(a—c)+ab(b—a)+bc(c—b), a3(b−c)+b2(c—a)+c2(a—b), and (a—b)(b—c)(c—a).

4. (a+b+c)(ab+ac+bc)—abc, and (a+b)(b+c)(c+a).

5. a(b—c)3+b(c—a)3+c(a—b)3, and (a—b)(b—c)(c—a)(a+b+c). 6. a(b+c)2+b(c+a)2+c(a+b)3-(a+b)(b+c)(c+a), and a(b+c—a)3+b(c+a—b)2+c(a+b—c)2+(a+b—c)(b+c—a)(c+a—b). 7. }{(a+b−c)(a+b)+(c+a−b)(a+c)+(b+c−a)(b+c)}. }{(a+b+c)2+(a+b−c)2+(a+c—b)3+(b+c—a)2}, and 2(ab+ac+bc)—(a+b−c)(a+c—b)—(b+c—a)(a+b−c)

-(a+c—b)(b+c—a).

8. {a(a—b)+b(b−c)+c(c—a)}(a+b+c), and
1{(a+b)3+(b+c)3+(c+a)3—3(a+b)(b+c)(c+a).

9. (a3+b3+c2)3+2(be+ac+ab)3—3(a2+b2+c3)(bc+ac+ab)3,
(a3—bc)3+(b3—ac)3+(c2—ab)3—3(a2—bc)(b2—ac)(c2—ab), and

(a2+2bc)3+(b3+2ac)3 +(c2+2ab)3—3(a3+2bc)(b3+2ac)(c2+2ab). 10. (a+b+c)(a+b−c)(a+c—b)(b+c—a), 4a2b3— { c2 —(a3+b3)}2, a2(b2+c3—a2)+b2(a3+c2—b3)+c2(a2+b3—c3),

4(a2b2+a3c2+b3c2)—(a2+b2+c2)3, (a2+b3)3 —(a3—b3)2 — (a3+b2—c2)3.

and 2a'b'+2aac2+2b3c2 — a1—¿a—ca.

XIV.

Verify the following equivalent expressions :

:

1. (1+x)(1+y3)—(1+x2)(1+y)=2(x-y)(1—xy). 2. (2x+3y)+(3x+2y)3 = 5(x+y) (7x2+11xy+7y3).

3. (a−b)(x − a)(x − b)+(b−c)(x−b)(x −c)+(c− a)(x — c)(x − a) =(a−b)(b−c)(c-a).

4. (x+b)(x+c)−(a+b+c)(x+b)+a2+ab+b2+3ax =(x+a)3-(a-b)c.

-

5. (xy—a2)2+(ay — bx)(ax — cy) = (ca — x2) (ab — y3)+(bc — a2)(xy — a2). 6. (a+b+c)(x+y+z)+(a+b−c)(x+y−x)+(b+c−a)(y+z−x)

+(c+a−b)(z+x-y)=4(ax+by+cz).

7. (x+y+z—xyz)2+(xy+yz+zx—1)2 = (1+x)2(1+y2)(1+x2). 8. x(y+z)2+y(x+z)2+z(x+y)*—4xyz= (y+z)(z+x)(x+y). 9. (1+xz)(1+yx)2 — {(1−xz)(1—yz)+2xyz} =4(x+y-xy)(xyz3+xyz3+z).

10. (xyz+x2y—y2z+z2x)2+(xyz+xy2+yz2—zx2)3

=(x2+y3)(y2+z3)(z2+x2).

11. (x2—1)(y2—1)(x2−1)+(x+yz)(y+xz)(z+xy)

=(xyz+1)(x2+y3+z2+2xyz—1).

12. (y2—zx)(z2—xy)+(z2—xy)(x2—yz)+(x2—yz)(y2—zx) —(x2—yz)3 — (y2—zx)2 — (z2—xy)2 = (x+y+z)(3xyz—x3—y3—¿3). 13. (x2+y)(+x2) (y—z)+(z2+y2)(x2+y3)(≈—x) +(x2+z3)(y2+z2)(x-y)=x^(y-z)+y' (z—x)+z+(x-y). 14. (y—z)5+(x−x)3+(x−y)3 =5(y-z)(x-x)(x—y){x2+y2+z2-yz-zx-xy}.

15. (y—z)°+(≈− x)° + (xy)° = 3(y — z)2(z — x)2 (x − y)3 +2(x2+y2+z3-xy-yz—xz)3.

XV.

1. If A=ax+by, and B=bx-ay, shew that aA3+bAB+cB2 = (a+b+c)b2x2+(a2—2ac+b2)bxy+a2cy3.

2. If A-a-bc, B-b-ac, C=c-ab; prove that (A3—BC)bc = (B2 — ▲ C)ac = ( C2—AB)ab = abc(a+b+c)(A+B+C). 3. If a+b+c=A, ab+ac+bc B; shew that

A+2B-3AB2 = (a3+b3+c3—3abc).

4. Given a+b+c+d=A, a+b-c-d= B, a—b+c-d=C, a-b-c+d=D;

shew that AB(A2+B2) = CD(C2+D3), if ab(a2+b2) = cd(c2+d2).

5. If a+b+c+d=0,A= bcd, B = cda, C= dab, D=abc; then shall BCD+CDA+DAB+ABC=0.

6. If bc-d=A, ca-e-B, ab-f2= C, ef-ad-D, fd-be = E, de-cf=F; then shall

AD+BE2+CF2 = ABC-2DEF—(Aa+eE+ƒF)3.

7. If X-ax+by+cz, Y=cx+by+az, and Z=bx+ay+cz, prove that (a+b+c—3abc)(x2+y3+x3-3xyz) = X3+Y+Z3-3XYZ.

8. If X=ax+cy+bz, Y=cx+by+az, Z=bx+ay+cz; shew that X+Y+Z3- YZ-ZX-XY

= (a2+b2+c2—bc-ac-ab)(x2+y2+z3—yz—zc—xy).

9. If the six equivalents, a=xX, b=yY, c=zZ, 2A=yZ+xY, 2B=zX+xZ, 2C=xY+yX, be simultaneously true, then shall a42+bB2+cC2 = 2ABC+abc.

XVI.

If 28=a+b+c, 203 = a2+b2+c3, and 203 = a+b3+; shew the truth of the following equivalents::

1. 82+(8—a)2+(8—b)2 +(8—c)2 = 203. 2. (8-a)+(8-b)+(8—c)3+3abc = 8. 3. (8-a)(8-b)(s—c) = 83—80-abc. 4. (b+c)8(8-a)+a(s—b)(s—c)—2bcs =(c+a)s(8—b)+b(s—c)(s—a)—2cas =(a+b)8(8—c)+c(s—a)(s—b)—2abs.

5. 8(8—b)(s—c)+s(s—c)(s—a)+s(s—a)(s—3) —(8—b)(8—c) (s—a) = abc.

6. {(s—a)(s—b)(s—c)}2+{8(s—b)(s—c)}2+{s(s—c)(s—a)}* +{8(8—a) (8—b)}+28038 (8—a) (s—b) (8—c) = a2b3c2.

7. (o2—a3) (o3—b3)+(σ2 —b2) (02—c2)+(o2—a3) (o2—c2)

=48(8-a) (8—b) (8—c).

8. (8—a)(8—b)+(8—b)(8−c)+(8—c) (8—a) — 83—o3.

=

9. (o3—a3)(s—a)+(03—b3) (8—b)+(03—c3) (8—c) = a*+b‘+c^—803.

XVII.

1. If bz-cy=p, cx-az-q, ay-bx=r; then ap+bq+cr=0. 2. If a=y+z-2x, b=x+x-2y, c=x+y-23; find the value of 2+c2+2bc-a2 in terms of x, y, z.

3. If 3x-p+2g+2r, 3y=2p-q+2r, 3z=2p+2q-r; shew that x2+y3+z2=p2+q3+r3, and xy+yz+xz=pq+pr+qr.

4. Ifp=b+c+d—a, q=a+c+d—b, r=a+b+d—c, s=a+b+c-d; shew that p(q+pr+ps)+q(a+qs+rs)=4(ab+ac+ad+bc+bd+cd). 5. If (x—y)z2 = c3, (y—z)x2= a3, (x—z)y2 — b3, (x—y)(y—z)(z—x) = 3abc, then shall a3+63+c—3abc = 0.

6. If 2a=x+, 2b=z+x, 2c=x+y, find the value of the expression a+b+c—2b3c23—2c2a3—2ab in terms of x, y, z, and express (x+y+z)(xy+xz+yz)—xyz in the form of factors involving a, b, c.

7. y3+z3+m(y+x) = x3 +x3+m(z+x)=x3+y3+m(x+y), x, y, z being unequal, then each expression is equal to 2xyz.

8. If a3=y+%, b2=x+z, c2=y+x, and 28=a+b+c, shew that s(s—a)(s—b)(s—c)=4(xy+xz+yz).

9. Shew that (a+b+c−d)(a+b+d—c)(a+c+d—b)(b+c+d−a) = 16(8—a)(s—b)(s—c)(s—d); if 28 = a+b+c+d.

10. If a+b+c=38, shew that (s—a)*+ (8—b)*+(8—0)" =2{(8-6)2(8—c)2 +(8—0)3 (s—a)2+(8—a)2(s—b)'}.

11. If x+y=p and xy=q, find x+y', x+y', '+y', &c., in terms

of p and q.

12. If a2+b2=c2, the product (a+b+c)(a+b−c)(a+c—b)(b+c—a) is equivalent to 4a2b3.

13. If xyz=a, y2-xzb, and x-xy-c; then shall x2+y3+z3-3xyz=ax+by+cz.

XVIII.

1. If x(1+y)=1 and y(1+2)=s, then -s=1+x+2x2+4x3+.... 2. If ay+bx=a, by-ax=b, then 2+y=1.

3. If xy=2(x+y-z), then shall (x-x)=yz.

4. If x+y+x-xyz=2, then shall (1-x)= (1-—xy)(1-xz).

5. If b(bx2+a3y) = a(ay2+b's), then shall bx+ay=ab, and ay = bx. 6. If (a+b-c-d)x = cd-ab, then (a+x)(b+x)=(c+x)(d+x). 7. If ax+by=1, then ab(x2+y2)+(a3+b2)xy+(a—b)(x—y)=1. 8. If (a-be)x+(b3—ca)y+(c3-ab)z=0 and x+y+z = 0, prove that ax+by+cz = 0.

9. If (a+bc)(1—a2)=(b+ac) (1-62), then a+b+c2+2abc1. 10. If x3+y3=z2, prove that (x3+z3)3y3+(x3—y3)3x3 = (y3+z3)3ï3. 11. If a+b+c=0, then 6(a+b+c3)=5(a3+b3+o3)(a2+b2+c2), 4(a2+b2+c2) = 7(a*+b+c‘)abc, and (a3+b3+c3)3 = 27a3b3c3.

12. If ax+by-c3=0, ay2+b2x-c-0 and x+y-c-0, then b2 = ac. 13. If (2a-3y)y = (z—x)2 and (2a—3z)z = (x—y)2, then shall x+y+z = a and (2a-3x)x = (y—x)3.

14. Ifa+b+c= 0, and a (by+cz-ax)=b(cz+ax-by) = c(ax+by—cz), then will x+y+z=0.

15. If (by-cx) = (b2—ac)(y3—cz), then shall

(bx—ay)2 = (b3—ac)(x2—az).

16. If a+b+c+d= 0, then shall a‘+b+c+d'+4abcd =2{(ab-cd)+(ac-bd)+(ad+bc)}.

17. If (yz-x)+(xx-y)2+xy-z)=(yz-x)(zx-y)(xy-z)+4, then x2+y3+z2 = xyz+4; and conversely.

XIX.

1. If a be greater than b, a"-b" is greater than nb"-1(a-b) but less than na-a-b).

2. Write the pth term of the quotient of aTM-bTM when divided by a"-b".

3. Show that a”—a" is divisible by a+1, when m―n is even; aTM+a* is divisible by a+1 when m―n is odd; and that a”—a" is divisible by a-1 when m―n is even or odd.

4. Prove that am-1 is always divisible by a"-1 and by "—1; and write the middle terms of the quotients when m is odd, and when n is even.

5. Determine the number of factors by which P-1 is divisible,

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