Page images
PDF
EPUB
[ocr errors][ocr errors]

when m, n, p are each odd and even numbers; and state the number of terms in each quotient.

6. Shew that 264 = 1+(1+2)(1+22)(1+2)(1+2)(1+2:0)(1+282).

7. Shew that a"-na-Ix+(n-1)a" is divisible by (x-a)', without performing the operation; and write the exact quotient of 2-52*x+40% divided by (x-a).

8. Prove that (an-1-3-1)(a*—6")(a*+1_8+1) is divisible by (a-6)(a-6)(a-63).

XX. 1. Shew by means of division if x +acy +y +bx+cy is resolvable into two rational factors, that abc = 62 +c.

2. Write the coefficient of x* in the product of t-ar tari-acta', ** + axta and x-ax+a'.

3. How many factors of the form x+a, and of the form x—b, are contained in the product w-11x +33.29-11x*—154x+120 ?

4. Divide x+px' +pu+1 and 2+px*+qx+qr+pr+1 by x+1 respectively, employing any artifice to save the trouble of formal division.

5. What is the criterion by which it is known that an Algebraical polynomial with integral coefficients, and arranged according to descending powers of x, is divisible (1) by x+1 and (2) by x-1? Shew that 2* +3.04 -20--2 is divisible both by x+1 and by :-1, and write the quotient.

6. Shew that x-px} +qx-rx+8 is exactly divisible by <-a, if al-pa+gai-ra+s=0.

7. Divide m'px' +qu-rx'+80-t by x-a as far as five terms of the quotient, and prove the correctness of the quotient by multiplying the divisor and quotient together and adding the remainder. Also write the quotient and remainder, (if any) when 20-8.3* +12.18-18.2° +200-30 is divided by x-4.

8. Shew when ao tayx+ayx +azdot .... +anx", can be divided by 1+x.

9. If x"+p,x*-!+px*-+ .. .. +Pn, be divisible by <-a, without remainder, shew that a" +p,a*-*+pa"-'+.... +p, is equal to

zero.

[ocr errors]

10. If ru, ru, rs

rn be the successive remainders when x"+pix*-+p-+... +pn is divided by x+a, n times in succession; then shall 2" +2,2"-? + 2x*-+.... tpm = (x+a)"+r.(z+a)*-' + rm1(x+a)*-' + ... trie Ex. 1+5.29 +64 +5.x+1= (+2)-3(x+2)+9(x+2)-9.

11. If there be a series of factors x+y, x+y-1, x+y-2, &c., and if the second factor be written in the forms (0–1)+y and +(y-1), then the product of the first two factors will take the form of

*(x-1)+2xy+y(y-1).

If the third factor be written in the forms (x-2)+y, (x-1)+(4-1), and x+(y-2); then the product of the first three factors takes the form of

(2-1)(x-2)+ 3xy(x-1)+ 3xy(y-1)+y(y-1)(y-2). Generalise this property.

XXI.

[ocr errors]

1. If a, b, c be in order of magnitude, ascertain when ab +ac+bc 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+4= 0, then aʼzy+b+xz+cʻxy will always be a positive quantity.

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

4. Show that a® +° is greater than 2ab, and (a+b)s than 4ab.
5. Is ab +ac+bc greater or less than a+63 +0% ?
6. Show that (a +6–c)+(ate-6)+(6+e-a)> ab +ac+be.

7. If a, b, c and a b + ab +ac+ar+6*c+bc-a-09--2abc be positive, then shall b+c>a, a+b>e and b+o>a.

8. If a>b, then (a® +62)" > 2ab(x*—ab+b).

9. If a> b and b>c, then (a+b+c)(ab+ac+bo) > 9abc and (a+b)(b+c)(c+a) > 8abc.

10. Show that abo is greater than (a+b-c)(a +(-6)0+b-a).

11. Shew that a +58 > a+b+ab and a +85>**+aobs; and generally that a*+*+*+">a"b" +a"b".

12. Is ao+a*b*+ab+89 always greater than (a +89)*?
13. Shew that a +28+ > 3abc, and a +84 +c* > abc(a+b+c).
14. Is (2*+y)(a'+b*)>(ax+by)* ?
15. Which is the greater (x+a)*(**+B) or (x+5)*(**+a") ?

16. If x= a +°, ya = c+d", then xy > ao+bd unless ad = be; and xy > ad+bc, unless ac = bd.

17. Shew that (a'+b+c)(x+y+z) is greater than (ax+by+cz). 18. Shew that (x+y+s) > 27xyz. 19. Shew that (1) (x+y)+(x+5-y)+(y+5-2)*><*+y+z.

(2) 28 thy' +s>${x*(y+x)+y*(x+x)+s*++y)}.

(3) (x+y-x)++-y)+(4+5-x) >x+y+x. 20. If x, y, s be real quantities in order of magnitude, then shall

(1) X(y-2)+y*(*—*)+*+(xy) be positive.
(2) (y-2)(s—2)+(5—X)(s—y)+(x−y)(y-2) be negative.
(3) 20(-Y)(---)+y(y->)(y-2) +8°(8—x)(-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., p. 30, &c. 7. Section I., pp. 7, 23. Section II., p. 22. 8. Section II., p. 28, &c. Section III., p. 8, &c.

II.

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+26+c. 2. -a+26-C. 3. a-c. 4. - 2a+c. 5. a-c. 6. 6+d. 7. 6-d. 8. a+b+c-d. 9. 5(a+b) – 15(c-d). 10. -2(a+c-d). 11. 96. 12. 10a - 2c.

IV. 1. 2ax. 2. 5a +b+13c. 3. 2a - 26+y. 4. (c +26-a)x+2by. 5. 262. 6. {(a+b-cha+(6+c-a)y+(a +c-b)z}.

[merged small][merged small][merged small][merged small][ocr errors]

VI. 1. Here a> 6 and 6>c..a>c. Hence a-c> , 6-c>0, a-c>0...2-C, 6-c, a-c are positive; and c-a, c-6, 6-a are negative.

2. See Section IV., Art. 6. 3. +(a-36+3c) and - (36-2-3c). 4. The sums and differences are ta-a, +a+b, ta-b, - a+b, –a-6, +6-6.

5. The aggregates are, ta+b+c, -a-6-6 +a-6-6, +6-a-c, +c-a-b, a+b-C, a+c-6, 6+c- a. 6. If x, y denote the two quantities, the quotient is

XY

(x+y)(x - y)" 7. ** +32x2 – 320x—1024. 8. The aggregate is 0. 9. (a?-ab-6)+(a? +ab+ba)y+(6: - ab-a?)z. 10. (a, +a, +a, - a „)x+ (a , +ag+a, -a,)y+(a, ta, +a.-a,)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, 66-5, 12c=5; a= }, 6-4, 6-t's, 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 +62 ; difference 4x – 2y – 52 ; sum of results 11x+3y +z, and the value is 1410.

5. The sum is 5x+13y +6z; the value is 6.3062.
6. The sum is 9(x+y+z), and the value 9.99.
8. The values are •584192 and .583808.
9. -2° +8x’y+ 7y +7yo.
10. px -(q+a)) + (- 6) * -(s+c)x: +(t-d)a? -(0+)!.
11. See Section IV., Art. 7, p. 6. Note.

RESULTS, HINTS, ETC., FOR THE EXERCISES ON

MULTIPLICATION AND DIVISION.

I. 2. See Section IV., Art. 7, p. 7. 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 : - 60a3b%cod: : +96068cds. 8. 5abx : - nc : 5ab’ds : - 12a2bc8. 9. 27a? + bax : -12a2b+ 24ab? : - ax: +a?? - asa:

- 180*62c? +24a3boc2 + 30a2b4c? : ma’x’yz – na’xy?z + pa’xyz' :

- a2z^^ 2 +b2x^^33 - coc?vozo. 10. 9a+2x : 4a+86 : 2? - ax +a? : 3a? – 4ab - 56: max- nay+paz : ac3+b^ – c2c2.

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.

[ocr errors]
[ocr errors]

II. The following are the products :1. 4x2 + 12xy +9y? : 4x2 - 9yo.: 4x? – 12xy +9yo. 2. x3 +70° +17x+35 : 6x8 – 19x?y+29xy? - 21y. 3. a* + a2x2 + x*. 4. 16a+ - 816*. 5. a4 +46*. 6. 35x5 – 49x4 + 1923 - 24x+64. 7. 26 - 12x4 – 576x? - 3600. 8. 26 - 5ax5 + 6a?.* + bas23 - 4a32 - 4a**. 9. 243x5 – ys. 10. 26 + 24 + x3 + x2 +1. 11. 98 +226 + 3a* + 2a? +1. 12. 1-7X+21x2 – 3508 + 40x+ - 2125 +70 - ?. 13. 2-6+72-4 - 64. 14. 2-4-y-4. 15. 6x-10 + 5x-'y-1-62-8y-. 16. 5x®y-4-12x*y-% +11.62 – 7y2 + 4x-?y- -"ye.

IIL The quotients are1. ** +63% + 112 +6. 2. 2? - x+1. 3. 16x® - 4xy + y'. 4. a^+2a+b+4a3b3 +8abo +166*. 5. 23 - 2+y^. 6. 3x8 – 2x2 - 52 - 3. 7. 4x02 - 5xy + 2y?. 8. 7:03 – 3x2y + 4xy? Bye. 9. 2a? - Bab + 56. 10. – 200* + 428 - 582 +102—20, and remainder + 49. 11. * - 4x3 + 6x2 - 4x +1. 12. 23 +° y + xy? +y'. 13. - 1+2-1. 14. ax-1-5-6-22. 15. 2x2 – 3xy +2./.

IV.

The quotients are1. a? - 2ab +63. 2. 2c? – (a – 26)x +(a? +382). 3. (x + y)2 + (c + y) 3xy +1. 4. a? +62 +62 - ab - 00 - - bc. 5. x2 + ax + b. 6. (a? + a +1)*; - (a +1). 7. a? +62 +c. 8. (a - b)x – (a - b)y. 9. x2 + y2 +22 +1. 10. a? - 2ab +36?. 11. (46 – 3c)? +(562 – 3c2 )x. 12. a* - 5a8b+5abs 64. 13. 2? – (a+b)x+ab. 14. a3-9x?. 15. 2° + sata?.

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

2(y-2)yz + (yo – z2)+x?(y z). 17. Proceed as in the last example. 18. a. 19. 2(a+b)..

V. 1. 81- 792 (81+79)(81 - 79) = 160 X 2=320. (1214)2 – (1187)2 = (1214+*)(1214 - 1187)=122x31=427.

2. {}(a+b)}-{f(a-6)}' =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. (as +63)2(a2 +62)3 = a12 +3a1062 +2a9b3 +3a8b4 +6a775 +2a6b6 +62567 +3a*68 +2a369 +3a2b10+812.

6. (c+a)2. 8. 2.05.2+x- 3. 9. (a? +ab +62)+3k(a+b)+3k.
10. – 216. 11. 2a- 3ab+402.
12. (mx+a)(nx+6)=mnx2 +(mb+na)x+ab

(mx a) (nx b)=MnX? (mb +na)x+ab
(mx+a)(nx—b)=mnxa +(mb - nabu-ab

(mx – a)(nx+b)=mnx? --(mb na)x - ab. As an example, take (3x+5)(5x+3). 13. (1) The quotient is ax+b, the remainder (c- aba)a +(d89), of which each

part must be equal to zero; or c-ab2 = 0, and d - 63 = 0, whence d=63

and c=cb? and ad=bc. (2) The quotient is ax? — aba+b, and the remainder (ab3 62)x+(0–68); of

which each part must vanish ; or abs 62=0, -63 = 0, whence 68 = c, and

ac-b? = 0, or b2 = ac. (3) After the quotient 23 +20x2 +3c2x+4c3 is found, the remainder is

- 5(9-c*)x+4(r -c), which must be equal to 0, and consequently 9-C4 = 0 and 7—65 = 0

..c* =q and c5 =r, ..020 =qs and c20 = 7?,:q5 =ph. 14. In the product make such substitutions as will make each of the three factors, identical to a ty.

« PreviousContinue »