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(a+b+c)-(a? +37+)
= 7(a+b)(b+c)(c+a){c++b+c+ab+ac+bc)+(a+b+c)abc}.

XI. Reduce to their simplest forms 1. (a+b)(+c)-(c+d)(d+a)-(a+c) (6-d). 2. a(a+b+c-d)+ba+b-c+d)+c(a−6+c+d)+d1-a+b+c+d). 3. (a+b)*+(a + c)* +(6+c)? +(a+d)+(6+d)* +(c+d)?. 4. (a +b+c+d)' +(@–6–0+d): +(a −6+0-d)?+(a+b-c-d). 5. {(a5c)a-(a+0–6)6+(6+ola+(a—c)c}(ab). 6. (a*—bc)(6—c)+(b? — ac)(-a)+(-ab)(ab). 7. (a26)(a +26)+(25 - 3c)(26+3c)+(3c – d)(3c+d). 8. (a+b+c)(a +b+d)+(a+c+d)(6+c+d)-(a+b+c+d). 9. (ab+ca)(co+do)+cd(a+b* 4). 10. {(ac+bd)”+(adbc)"}.{(ac+bd) - (ad + bc)"}. 11. 4{(a* — b*)cd +(c d?)ab}+{(a? — 6^)(c* d) -4abcd}. 12. (ai-670-d)' +2{(ab - cd)' +(bc +ad).-(ac - bd;}. 13. (a+b)* +(6+c)* +(c+d)* +(d+a): +(a +c)* +(6+d)". 14. (a8)8+(a+b)8+3{(ab)(a+b)+(a+b)*(ab)}. 15. a[b+c)' +(a+c)+c(a+b)2-((a+b)—B*(a+c) - a'(b+c). 16. (a+b+c)-(a+b-0)-(a+o-6):-(6+c-a)'.

17. (a +b+c+d)*-(a* +b+c+d) -(a+b+d)*-(a +c+d)*- (a+b+c)-(6+c+d). +(2+5)*+(6+0)* +(c+d)*+(a+d)*+(6+d)* +(a + c)'.

18. (6*6* + a*d*)(6-c)(a-d)+(c*a? +6*d*)(c-a)(6-d) +(a*b* +c+d)(a - b)(c-d).

19. (a+b+c+d)–(6+c+d)–(a +c+d)-(a +b+d)" - (a+b+c") + (6+c)+(a+o)+(a+b)+(a+d)*+(6+d)*+(c+d)" – (a* +60 +c+d).

XII. Verify the following expressions : 1. 30ab - (9a - 86)(5a +26)-(46 - 3a (158+46) = 4ab. 2. (ab+ac+bc)-(a*b* +a'c+ 6*c*) = 2(a+b+c)abc.

3. ab+c)(69+03 – a)+(c + a)(co+a— 64)+c(a+b)(a’ +62–c) = 2abc(a+b+c).

4. (a +b+c)-(a+b)-(6+)*-(+a)* + at +64 + c = 12abc(a +b+c).

5. a'(b+c-a)* +6*c+a-6)*+ (a +6-c)* + abc(a+82 +co) + (a* +6 +°—bc- ca ab)(6+0-a)(c+a-b)(a +6–c)=2abc(bc +ca +ab).

6. (a+b-2c) +(6+c-2a)+(c+a-26)* = 3(a+6-20) (6+c-2a) (a +(-26).

7. 2(a+b+c)-(a+b)-(6+c)-(c+a)'+3abo = 3(ab+bo+ ac)(a+b+c).

8. 8(a+b+c)8—(a+b)8—(b+c)2-(c+a)) = 3(2a+b+c)(a+26+c)(a +6+2c).

9. (a+b+c)*+a+68+8=(a+b)+(b+c)+(c+a)* +6abc.
10. (a-)+(b-c)+(-a)=3(ab)(-)(-a).
11. {(a−b)? +(6c)+(c—a)?} = 2{(ab)*+(6—c)+(c—a)}.

12. {(ab)*+(6—c)+(-a)?;8—54(a-6) (6c)"(c—a) = 2(a +6—2c)?(b+c-2a)"(c+a-26).

13. (ab)*+(6—c)+(c—a)* = 2{(ab)(c—a)'+(a−b)'(6c)? +(-c)'(c-a)"} = 2(a+b+-ab-bc-ac).

14. (1+-a'+ab+bc+ca)(-5)+(+a'-+ab+bc+ca)(a'-) +(a+62—c? + ab + bc+ca) = (a? —B*)(62—c*)(0-a?).

15. a'(b+c): +6*(c+a): +o(a+b)* +abc(a+b+c) +(a’ +62+c)(bc+ca +ab) = (a+b+c)(6+c)(c+a)(a+b).

16. {2bc(a6)-(B*+-—a*)(a-c)}'+(a-c)(b+c-a)(c+a-3) (a+bc)(a+b+c) = 4abc{abc-(6+c—a)(c+a—b)(a +b-c)}.

17. (b+c)^+8°(d+c)+++(a+b)^= 25**(a+b)(a+c) +2c*a*(b+c)'(b+a)' +2a+b'(a+c)'(c+)-16a+b+c+(bc+ac+ab).

18. (a+b)2(6+c)?(c+a)+2a2b°co —a'(b+c).—6*c+a)-c(a+b): =2(ab + bc+ca).

XIII.
Shew the identity in value of the following sets of expressions :

1. (a-b)+(-a)(c-b), (-c)' +(a−b)(a-c) and (c—a)* +(6—a)(6—c).

2. (a+b)' +(+c)'+(c+a)-(a'+b? +ca), 3(a +b+c)-{(a−b)*+(6—c)+(c—a)"}, and }{a—)*+(6—c)?+(c—a)+6(ab+ac+bc)}.

3. aca-c)+ab(b--a) +bec-), a'(b-c)+(c-a)+(a-6), and (ab)(6c)(c-a).

4. (a+b+c)(ab+ac+bc)-abc, and (a+b)(b+c)(c+a). 5. a(6-0)+b(c—a) +ca—b), and (ab)(6—c)(c—a)(a+b+c).

6. a(b+c)'+b(c+a)? +c(a+b)–(a+b)(6+c)(c+a), and a(btc-a)+(c+a-6)+((a+b—c)+(a+b—c)(6+c—a)(c+ab). 7. }{(a +6c)(a+b)+(c+a-b)(a +c)+(6+c-a)(6+c)}.

*{(a+b+c)'+(a+b—c)?+(a+o-)+(6+c—a)?}, and 2(ab+ac+bc)-(a+b-c)(a +c-6)-(6+c-a)(a+b-c) ー(a+c-b)(b+c-a). 8. {a(a-6)+(6-0)+cc-a)}(a+b+c), and

{(a+b)*+b+c)+(c+a))-3(a+b)(b+c)(c+a). 9. (a+b+ca)+2(bc +ac+able—3(a+62 +c*)(bc+ac+ab)',

(a*—bc)* +(—ac)* +(c'— ab)3—3(a'bc)(b? ac)(co —ab), and

(a? +2bc)*+(6°+2ac)* +(6+2ab)*—3(a* +2bc)(6*+2ac)(c +2ab). 10. (a+b+c)(a+b-c)(a +(-6)(6+c-a), 4a’b?—10—(a +6+)}},

a*(b*+*—a*)+B*(a’ +0—6*)+c(a’+b?—c),

4(ab*+a'c' +boc*)-(a'+b+c)', (a? +69)3—(a—62)-(a' +6?_co)?. and 2ao39+2a'd'+26* -a -o

XIV. Verify the following equivalent expressions : 1. (1+x)?(1+y)-(1+x^)(1+y): = 2(x−y)(1-xy). 2. (2x+3y)+(3x+2y): = 5(x+y)(7x? +11xy +7y?).

3. (a−b)(x – a)(x-6)+(6—c)(x-6)(x –c)+(c-a)(x-2)(x-a) = (2-6)(6-c)(c-a).

4. (x+3)(x+c)-(a+b+c)(x+6)+a' + ab +62+3ax = (x+a)-(ab)c.

5. (xy a?): +(ay bx)(ax – cy) = (ca x?)(ab yo)+(be - a)(xy — a*).

6. (a+b+c)(x+y+z)+(a+b-c)(x+y-x)+(6+c-a)(y+2-x) +(c+a-6)(3+x- y) = 4(ax+by+cz).

7. (x+y+z-xyz)"+(xy+ya+z+-1) = (1+x)*(1+y?)(1+z2). 8. x(y+z)'+y(x+3)+ (x+y)-4xyz = (y+z)(z+x)(x+y).

9. (1+xz)?(1+ys).— {(1-x3)(1–43)+2xyz}? = 4(x+y-xy)(xyz'+xyzo+s).

10. (xyz+xʻy-y:z+z+x)+(xyz+xy?+yz? — 2xo) = (x*+y*)(y'+*)(zo+x?).

11. (2-1)(yo— 1)(z:— 1)+(x+yz)(y+xz)(3+xy) =(xyz+1)(2+y+"+2xyz-1).

12. (yo—2x)(z?—xy)+(z'—xy)(x? —yz)+(x*—yz)(y2—42) -(x-yz)-(yo-zr)-(:'_xy) = (x+y+z)(3.xyz—*—y:—-").

13. (x+y)(='+x)(y-2)+(zo+y)(x+y)(:-*) +(x2+x")(y* +22)(-y)=x*(y-7)+y*%-x)+**(x−y).

14. (y-2)+(3^2)+(x-y)" = 5(y-2)(1-x)(-y){+*+y+z-y-20-xy).

15. (y-)+(-x)+(x-y)' = 3(y-2)*(2-x)'(x - y)" +2(x+y*+– xy - Yo— xz)*.

XV. 1. If A = ax+by, and B=bu-ay, shew that aA+AB+B=(a+b+c)b*z? +(a-2ac+64)bxy taʼcy.

2. If A = a-bc, B = b - ac, C = (-ab; prove that (A-BC)bc = (B-AC)ac =((?-AB)ab=abc(a+b+c)(A+B+C).

3. If a'+62+c = A, ab+ac+bc=B; shew that 4+2B-3A B = (a +58 +-3abc).

4. Given a+b+c+d= A, a+b-o-d=B, a-6+c-d-C, 4-6-c+d=D; show that AB(A + B) = CD(C++D), if ab(a' +34) = cd(c+d).

5. If a+b+c+d=0,4 = bed, B=cda, C = dab, D=abc; then shall BCD+CD4+DAB+ABC=0.

6. If bc-di= A, ca-co=B, ab-fo= C, e-ad=D, fd-be= E, de-ef=F; then shall AD+BE+CF?= ABC-2DEF-(Aa+E+fF).

7. If X= ax+by+cz, Y=cc+by+az, and Z=bx+ay+cz, prove that (a +68+0-3abc)(x+y+4—3xyz)= X+Y*+2-3XYZ.

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8. If X= ax+cy+b3, Y = cx+by+az, 2=bx+oy+cz; show that X+Y' +2'-Y2-ZX-XY

(d+do+co-be-ac-ab)(zo+v+s-y%-xg).

9. If the six equivalents, a=*X, b=yY, c= 2, 2A = y2+:Y, 2 B= X+x2, 2C=xY+yX, be simultaneously true, then shall a 4+B+CP= 2 ABC+abc.

XVI. If 28 = a +b+c, 20-a'+62+, and 20%= a +18+; shew tho truth of the following equivalents :

1. 3+(8—a)+(8—6)* +(8—c)? = 20%. 2. (8—a): +(8—6)8+(8-0)* +3abc = . 3. (8—a)(8—6)(8—c)=88-80-abc.

4. (b+c):(8-a) +a(8-8)(8-0)-2bc3 =(c+a):(8-5)+(8-c)(8-a)-2cas = (a+b)8(8-0)+c8-a (8-6-2abs.

5. 8(8-8)(8-0)+8(8—c)(8—a) +-8(8a)(8—) -(8-6)(8—c)(8—a)=abc.

6. {(8—a)(8—6)(8—c)}'+{818—6)(8—c)}'+ {818—c)(8—a)}" +(88-a)(8-6)}'+280*8(8-a) (8-8)(8-0) = a*b*c.

7. (o?—a?)(04—)+(93–69)(08—c)+(p*—a)(08-) = 48(8—a)(8—6)(8—c).

8. (8—a)(8—6)+(846)(8—c)+(8—c)(8—a)=3*—o?. 9. (os—a)(8—a)+(00—60)(8—6)+(-)(8-0) = a* +66+6—80°.

XVII. 1. If bx—cy =p, cướaz=q, ay-bx =r; then ap+bq+cr=0.

2. If a=y+8-2x, b=s+x-2y, c=x+y-22; find the value of 2+0+2bc-a in terms of x, y, %.

3. If 3x = -p+29+2r, 3y = 2p-9+2r, 3x = 2p+29-r; shew that 2°+y+z=p+q+ge, and xy+ya+#2 =Pq+pr+qr.

* 4. If p=b+c+d-a,q=a+c+d-b, r=a+b+8-C, 8=a+b+c-d; shew that pla+pr+p8)+9(2+98 +rs) = 4(ab+ac+ad+bc+6d+cd).

5. If (x,y)2 = (', (y-2).* = a, (x-2)y* = *, (3-y)(y-2)(2-) = 3abc, then shall a++08-3abc = 0.

6. If 2a = x+8, 2b = st*, 2c = x+y, find the value of the expression a*+8+4—20°C—2c*a*—2aobo in terms of l, y, , and express (x+y+x)(xy+23+yz)—xys in the form of factors involving a, b, c.

7. y8+$8+m(y+x) = 5+28+m(x+x) = +y+m(x+y), x, y, s being unequal, then each expression is equal to 2xyz.

8. If a'=y+s, 6 = x+s, ry+s, and 28 = a +6+c, shew that 8(8-a)(846)(8—c) = 4(xy+23+yz).

9. Shew that (a +b+c-d)(a+b+d-c)(a +0+2-8)(6+c+d-a) = 16(8—a)(8-8)(8-c)(0-d); if 28 = a +b+c+d.

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

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

12. If a+b = c', the product (a+b+c)(a+b-c)(a +(-)(6+c-a) is equivalent to 4a6*.

13. If x-Ys = a, yi-32=b, and -=0; then shall +y+-3xyz = ax +by+cz.

XVIII.

1. If x(1+y)= 1 and y(1+x) = x, then —8=1+x+2.ro +420 + ....
2. If ay+bx = a, by-ax = 6, then ** +y - 1.
3. If x'y = 2(x+y-x)', then shall (<->)* =yz.
4. If x+y+:-xyz = 2, then shall (1-2)' = (1-xy)(1-x2).
5. If (bx*+a*y)= a(ay' +6+x), then shall br+ay = ab, and ay = bx.
6. If (a+b-c-d)x = cd-ab, then (a +x)(6+x)=(c+x)(d+x).
7. If ax+by = 1, then ab(2*+y)+(a*+6*)xy +(a-6)(x-y)= 1.

8. If (arbe)x+(6—ca)y+(c—ab)3 = 0 and x+y+% =0, prove that ax+by+c%= 0.

9. If (a+be)'(1—a)=(6+ac)'(1 —b), then a +b+c+2abc == 1. 10. If x+y8=z?, prove that (x8+zo)*y$ +(.x—ys)*8 = (y8 +39)**.

11. If a+b+c=0, then 6(a' +65 +0%) = 5(a+b+c)(a +b+c), 4a++37+) = 7(a+b+c)abc, and (a++)* = 27a*b**.

12. If ax+by-= 0, ay +6*x-08-0 and x+y-c=0, then b = ac.

13. If (2a-3y)y=(2-x) and (2a-32)2 = (x-y), then shall x+y+z = a and (20—3x)x =(y-2).

14. If a+b+c= 0, and a (by+cz-ax)=b(cz tax-by) = c(ax+by-cz), then will x+y+s=0.

15. If (by-cx) =(B2- ac)(y-cz), then shall (bx-ay) = (b1-ac)(-az).

16. If a+b+c+d=0, then shall a +8+c+d'+4abcd = 2{(ab-cd)' +(ac-bd)' +(ad+bc)"}.

17. If (y-2)*+(2x−y)*+xyz)= (y3—X)(2x−y)(xy%)+4, then x+y+z* = xyz +4; and conversely.

XIX.

1. If a be greater than b, ar-6" is greater than non-a-b) but less than nan-a-b).

2. Write the pth term of the quotient of ama — fm* when divided by a" — 6m.

3. Show that a"-a" is divisible by a+1, when m-n is even; a" ta* is divisible by a +1 when m-n is odd; and that a" -a" is divisible by 4-1 when m-n is even or odd.

4. Prove that am-1 is always divisible by a"-1 and by 3-1; and write the middle terms of the quotients when m is odd, and when n is over.

5. Determine the number of factors by which 2MAP – 1 is divisible,

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