If the third factor be written in the forms (x-2)+y, (x−1)+(y-1), and x+(y-2); then the product of the first three factors takes the form of

x(x−1)(x−2)+3xy(x−1)+3xy(y-1)+y(y−1)(y—2).

Generalise this property.

XXI.

1. If a, b, c be in order of magnitude, ascertain when ab+ac+be is greater or less than each of the quantities a(a+b+c), b(a+b+c) or c(a+b+c).

2. If a+b+c=0 and x+y+x=0, then a3zy+b2xz+cxy will always be a positive quantity.

3. If a, b, c be three unequal positive numbers in order of magnitude, and x, y, z other three; it is impossible that ay-bx, bz-cy, and cx-az can be all positive.

4. Shew that a2+b2 is greater than 2ab, and (a+b)3 than 4ab.

5. Is ab+ac+be greater or less than a+b+c2?

6. Show that (a+b−c)2+(a+c-b)2+(b+c-a)> ab+ac+be. 7. If a, b, c and a2b+ab3+a2c+ac2+b2c+bc3—a3—b3—c3—2abc be positive, then shall b+c>a, a+b>c and b+c>a.

8. If a>b, then (a2+b2)2 > 2ab(x2—ab+b3).

9. If ab and b>c, then (a+b+c)(ab+ac+be) > 9abc

and (a+b)(b+c)(c+a)>8abc.

10. Shew that abc is greater than (a+b-c)(a+c—b)(o+b—a). 11. Show that a3+b3>a3b+ab2 and a3+b3>a3b2+a2b3; and generally that aTM+*+b+">amb"+a*bm.

12. Is ao+a1b3+a2b1+6o always greater than (a3+b3)3 ?

13. Shew that a+b+c> 3abc, and a+b+c> abc(a+b+c).

14. Is (x2+y2)(a2+b2)>(ax+by)?

15. Which is the greater (x+a)(x2+b2) or (x+b)2(x2+a3) ? 16. If x3 = a2+b3, y3 = c2+ď2, then xy > ac+bd unless ad = bc; and xy> ad+be, unless ac-bd.

17. Shew that (a2+b2+c2)(x2+y2+z2) is greater than (ax+by+cz)3. 18. Shew that (x+y+z)3 > 27xyz.

19. Shew that (1) (x+y-x)+(x+z—y)2+(y+z—x)3>x2+y2+z2. (2) x3+y3+x3>} {x2(y+s)+y2(x+z)+z2(x+y)}.

(3) (x+y-z)+(x+%—y)3+(y+z−x)3>x3+y3+z3. 20. If x, y, z be real quantities in order of magnitude, then shall (1) x3(y—z)+y2(z—x)+z2(x—y) be positive.

(2) (y—z)(s—x)+(x−x)(z—y)+(x—y)(y—z) be negative.
(3) x3(x—y)(x—s)+y3(y—s)(y—x)+x3(z—x)(z—y) be positive.

RESULTS, HINTS, ETC., FOR THE EXERCISES

ON ADDITION AND SUBTRACTION.

I.

1. See Section I., p. 12. 2. Section II., pp. 1, 4, 5. 3. Section II., p. 2, &c. 4. Section 1., pp. 5-8. 5. Section II., p. 16. 6. Section II., I., pp. 7, 23. Section II., p. 22. 8. Section II., p. 28, &c.

II.

p. 30, &c. 7. Section Section III., p. 8, &c.

1. See Section IV., Art. 3, p. 2: Art. 9, p. 10: and Art. 7, p. 6.

2. Two numbers are equal when each of them consists of the same number of units. When geometrical magnitudes are equal, see School Euclid, Notes on the Axioms, p. 46.

3. The sum is +12a; the difference +7a.
4. The sum is -12a; the difference -7a.
5. The sum is +2a; the difference +7a.
6. The sum is +2a; the difference +26.
8. See Section IV., Art. 9, p. 17.
9. 8. 10. 2(m+n+p).

11. 2ma the sum; 2na the difference.
12. See Section IV., Art. 7, p. 5.

III.

1. 3a+2b+c. 2. —a+2b-c. 3. a-c. 4.-2a+c. 5. a-c. 6. b+d. 7. b-d. 8. a+b+c-d. 9. 5(a+b)−15(c–d). 10. −2(a+c−d). 11. 9b. 12. 10a-2c.

IV.

1. 2ax. 2. 5a+b+13c. 3. 2a-2b+y. 4. (c+2b-a)x+2by.

5. 2bx. 6. {(a+b−c)x+(b+c−a)y+(a+c—b)z}.

V.

1. 8x-y. 2. -2(x-y-z). 3. -(x+y+z). 4. x-5y. 5. 53x-2y-4%.

6. 3x+7y+11z.

VI.

1. Here ab and b>c...a> c. Hence a-c>0, b-c>0, a-c>0...a-c b-c, a-c are positive; and c-a, c-b, b-a are negative.

2. See Section IV., Art. 6. 3. +(a-3b+3c) and (3b-a-3c).

4. The sums and differences are +a-a, +a+b, +a−b, −a+b, −a−b, +b-b. 5. The aggregates are, +a+b+c, -a-b-c, +a-b-c, +b-a-c, +c-a-b, a+b-c, a+c-b, b+c-a.

6. If x, y denote the two quantities, the quotient is

7. x+32x2-320x-1024. 8. The aggregate is 0. 9. (a2-ab-b2)x+(a2+ab+b2)y+(ba—ab—a2)z.

xy

(x + y)(x − y)°

10. (a,+a,+a ̧ − α)x+(α2+a2+ α ̧ − α ̧) y + (α ̧+a ̧+α ̧—α ̧)z. 11. See Section IV., Art. 3, p. 2; Art. 9, p. 9

VII.

1. The sum is 2(a+b+c), and since 3a=5, 6b-5, 12c=5; a=§, b=f, c=√1⁄2, the

value is 5.

2. Apply Axiom 1.

3. The simplest form is 9x-14, and the least integral value of x, which renders the result negative, is unity.

4. Sum 7x+5y+6z; difference 4x-2y-5z; sum of results 11x+3y+z, and the value is 1410.

5. The sum is 5x+13y+6z; the value is 6.306x.

6. The sum is 9(x+y+z), and the value 9.99.

8. The values are 584192 and 583808.

9. -x3+8x3y+7xy2+7y3.

10. px − (q+a)x3 + (r− b)x+−(s+c)x3 +(t−d)x2 — (v + c) L.

11. See Section IV., Art. 7, p. 6. Note.

RESULTS, HINTS, ETC., FOR THE EXERCISES ON
MULTIPLICATION AND DIVISION.

2. See Section IV., Art. 7, p. 7.

I.

3. See Section IV., Art. 3, p. 4; Art. 5, p. 5.

4. See Section IV., Art. 9, p. 9; Art. 10, pp. 14, 15.

5. See Section IV., Art. 12, pp. 15, 16.

6. See Section IV., Art. 14, pp. 24, 25.

7. 20abcdxy: -mnabc: -60a3b3c3d3: +96a3b3c3d3.

8. 5abx: - nc: 5ab2d3: -12a2bc3.

9. 27a2 + 6ax: -12a2b+24ab2: - ax3 +a2x2 −a3x :

- 18a4b2c2+24a3b3c2 +30a2b1c2: ma2x2yz -— na3xy2z+pa3xyz" :

- a3x1y3z2+b2x2y1z2 — c2x2y2z1.

10. 9a+2x: 4a+8b: x2 − ax+a2: 3a2 - 4ab-5b3: max-nay+paz :

— a2x2+b2y2 — c2z2.

11. See Section IV., Art. 12., pp. 17, 18.

12. See Notes on Euclid II., 1-10, of the Editor's Edition of Euclid's Elements. 13. See Section X., Art. 13, p. 20.

The following are the products :

II.

1. 4x2+12xy+9y2: 4x2 - 9y3.: 4x2 - 12xy +9y3.

2. x3+7x2+17x+35: 6x3-19x2y+29xy2 - 21y3.

3. a++a2x2+x+. 4. 16a+ - 81b*. 5. a1+4ba.

6. 35x3- 49x1 +19x3 − 24x+64. 7. x-12x1 - 576x2 - 3600.

8. x-5ax +6a2x2+6α3x3-4a3x2-4a4x.

9. 243x-y3. 10. x + x + x3 + x2 + 1.

11. aa + 2a+3a1 +2a2 +1. 12. 1-7x+21x2 - 35x3 + 40x1 − 21x3 +7x® −x2.

13. x-6+7x-4-64. 14. 24-y-4. 15. 6x-10+5x−9y-1 − 6x—3y—2.

16. 5xy-4-12x1y−2 + 11x2 -7y2+4x-2y1 —x—4y®.

IIL

The quotients are

1. x2+6x2+11x+6. 2. x2-x+1. 3. 16x2 - 4xy+y'.
4. a1+2ab+4aab2 +8ab3 +16b+. 5. x2-xy+y2.
6. 3x3- 2x2 -5x-3. 7. 4x2 - 5xy+2y2.

8. 7x3-3x2y+4xy2 - 8y3. 9. 2a2-3ab+5b2.

10. x3-2x+4x3-5x2+10x-20, and remainder + 49.
11. x - 4x3+6x2-4x+1. 12. x3 +x3y+xy2+y3.

13. x-1+x-1.

14. ax-1-b-b-2x.

15. 2x2-3xy+2.ƒ3.

IV.

The quotients are—

1. a2 - 2ab+b2. 2. x2 - (α- 2b)x+(a2+3b2). 3. (x+y)2+(x + y) − 3xy +1. 4. a2+b2+c2 -ab-ac-bc. 5. x2+ax+b. 6. (a3+a+1)x− (a+1).

7. a2+b2+c2. 8. (a - b)x− (a−b)y. 9. x2+y2+z2+1. 10. a2 – 2ab+3b2. 11. (4b-3c)x2 +(5b2-3c2)x. 12. a1 - 5a3b+5ab3 - b1. 13. x2 − (a+b)x+ab. 14. a3-9x2. 15. x2+xa+a3.

16. Arrange the dividend in ascending powers of x, making (y+≈) −x the divisor; perform the division and verify the result. The quotient is

2(y − z)yz+x(y3 − z2)+x2 (y − z).

17. Proceed as in the last example.

18. a. 19. 2(a+b)x.

V.

1. 812-792 (81+79) (81 −79)=160×2=320.

(1214)2 - (1184)2 =(1214+4)(1214-1184)=122×31-427.

2. {(a+b)}-{(a-b)}=ab. The product of any two numbers is equal to the square of half their difference subtracted from the square of half their sum. 3. x2+1. 4. x=9.

5. (a3+b3)2(a2+b2)3=a12+3a1ob2+2a3b3+3a3b1+6a ̄b3+2a®b®+6a3b2

+3a+b+2a3b+3a2b10+b12.

6. (x+a)2. 8. 2·05x2+x-3. 9. (a2+ab+b2)+3k(a+b)+3k2. 10.216. 11. 2a2-3ab+462.

12. (mx+a)(nx+b) = mnx2+(mb+na)x+ab

(mx − a)(nx − b)=mnx2 − (mb+na)x+ab
(mx+a)(nx-b) = mnx2 + (mb −na)x-ab
(mx− a)(nx+b) = mnx2 — (mb — na)x− ab.

As an example, take (3x+5) (5x±3).

13. (1) The quotient is ax+b, the remainder (c- ab2)a+(d− b3), of which each part must be equal to zero; or c-ab2=0, and d-b30, whence d=b3, and c=cb2 and ad=bc.

(2) The quotient is ax2-abx+b, and the remainder (ab3 - b2)x+(c−b3); of which each part must vanish ; or ab3 —b2 = 0, c—b3 =0, whence b3 =c, and ac-b2=0, or b2 = ac.

(3) After the quotient x+2cx2+3c2x+4cs is found, the remainder is −5(q−c1)x+4(r-c3), which must be equal to 0, and consequently q-c0 and r—c3 =0

...c4=q and c3 =r, .'.c2o =q3 and c2o=r2.. q5 =p4. 14. In the product make such substitutions as will make each of the three factors, identical to x+y.

VI.

1. x-4x3-9x2+13x+18.

2. x+25x4+254x3+1307x2+3475x+3552,

and x-5x+14x3-25x2+23x-6.

3. The product is x2+(a+b+c+d)x3+(ab+ac+ad+be+bd+cd)x +(abc+abd+acd+bed)x+abcd.

The coefficient of x in the required product is 2.6.10+2.6.14+2.10.14+6.10.14=1408.

4. —4x2y(x-y).

5. The differences are each 16x(x2+1).

6. By aid of a3 — b2 = (a+b)(a−b); the dividend is equal to 5(x2+x+1)(x2 - 13x-1).

7. x2 + xy + y2. 8. x2-8xy-15y2.

9. The coefficient of x is +193, and of x is +83.

10. 4x2+8xy+4y2 —6xz+3yz+9z2.

This may be found by inspection with the aid of the quotient (a3 +b3)÷(a+b).

VII.

The products under 1, 2, 3 may be inferred from the form (a+b)(a - b) —aa — b2. 1. a2+b2 - c2+2ab; c2 - a2 − b2+2ab ; a2 + c2 − b2 +2ac; b2+c2-a2+2bc; a2 - b2 - c2+2bc; b2 - a2 - ca + 2ac.

2. a2+b2+c2 - d2+2ab+2ac+2bc; a2+d2 − b2 − c2 + 2ad +2bc;

a2 + d2 - b2 - c2 - 2ad+2bc; a2 + c2 − b2 — d2+2ac2bd;

d2 a2 - b2 -- c2+2ab+2ac - 2bc; b2+d2 - a2 - c2 - 2bd-2ac.

3. a1+a2b2+ba. 4. 1−(a+b+c)x+(ab+ac+bc)x2 - abcx3.

VIII.

1. (a−c)(a+c−2). 2. (a+b)(b+c)(c+a). 3. (a+b)(b+c)(c+a).

4. (a+b+c-d) (a − b − c − d).

5. (a+b+c−d)(a+b+d−c) (c+d+a−b) (c+d+b− a).

6. (x+1)(x − 2)(x − 3)(x − 4).

7. (a+b+c+d) (a + d − b −c)(b+c-a-d) (b-c-a+d).

8. (x+a)(x+b)(x2 - ax+a2). 9. x(x-2α)(x2 - ax+a2)(x2 - 3ax+3a2). 10. (1+y)2(1 − x)2(1 + x)2.

11. ((a+1)x-(b−1)y}(x2 - xy + y2).

12. (abc+bcd+cad+abd)2 = {ac(b+d)+bd(a+c)
+ c)} 2

=a2c2(b+d)2+2abcd(b+d)(a+c)+b2d2 (a+c)2,

and {(b+d)+(a+c)} 2 abcd=abcd(b+d)2 +2abcd(b+d) (a+c) + abcd(a+c)2, ...(abe+bcd+cad + abd)2 - (a+b+c+d)2 abcd

≈ac(b+d)2 (ac — bd) — bd(a+c)2 (ac — bd)=(ac − bd) { ac(b+d)2 − bd(a+c)'} = (ac—bd)(ab - cd) (bc — ad).

1. (a+b)(a+c) = a2 + (b+c)a+bc

(d+b)(d+c)=d2+(b+c)d+bc

IX.

•*. (a+b)(a+c) − (d+b)(d+c) = (a2 − d2) + (b + c)(a–d), which is divisible by a-d. 2. The dividend can be arranged thus: c(a - b2) — ab(a - b) — c2(a - b) which is obviously divisible by a-b.

3. The dividend is the difference of two squares, which may be expressed as the product of the sum and difference of the quantities. The quotient is

(a−b)x+(b −c)y + (c− α)z.