53. We add a few harder cases worked out in full. x2+2xα+2α2) x2+4a1 - 2x3α- 4x2a2- 4xα3 2x a2+4xα3+4a Example 2. Divide a3+b3+c3 - 3abc by a+b+c. a+b+c) a3-3abc + b3+ c3 (a2 - ab- ac+b2-bc+c2 NOTE. In the above example the dividend and successive remainders are arranged in descending powers of a. The result of this division is very important and will be referred to later. 54. When the coefficients are fractional the ordinary process may still be employed. Example, Divide 13+xy+y3 by x+y. x+y)‡x2+71⁄2xy2 + √»y3 (}x2 − {xy+{y2 4x3 + x2y In the examples given hitherto the divisor has been exactly contained in the dividend. When the division is not exact the work should be carried on until the remainder is of lower dimensions [Art. 10] than the divisor. Divide EXAMPLES VI. c. 1. x3-x2-9x-12 by x2+3x+3. 2. 2y3-3y2 - 6y – 1 by 2y2 – 5y – 1. 3. 6m3-m2-14m+3 by 3m2 + 4m − 1. 4. 6a5-13a4+4a3+3a2 by 3a3 – 2a2 — a. 5. x4+x3+7x2-6x+8 by x2+2x+8. 6. a4-a3. ·8a2+12a − 9 by a2+2a−3. 7. a*+6a3+13a2+12a+4 by a2+3a+2. 8. 2x4x3+4x2+7x+1 by x2−x+3. 9. x5-5x+9x3 − 6x2 −x+2 by x2 - 3x + 2. 10. x2-4x1+3x2+3x2 −3x + 2 by x2 − x − 2. 11. 30x2+11x3- 82x2 - 5x+3 by 2x-4+3x2. 12. 30y+9-71y3+28y1 — 35y2 by 4y2 – 13y+6. 13. 6k5-15/4+4k3+7k2 −7k+2 by 3k3 − k +1. 14. 15+2m2-31m+9m2+4m3 +m5 by 3 - 2m – m2. 15. 2x3- 8x+x1+12−7x2 by x2+2−3x. 16. x5-2x4-4x3 +19x2 by x3 −7x+5. 17. 192-x+128x+4x2 - 8x3 by 16-x2. 18. 14x+45x3y+78x2y2+45.xy3+14y+ by 2x2+5xy+7y2. 19. xxy+x3y2-x3+x2-y3 by x3-x-y. 20. x2+x1y-x3y2+x3 − 2xy2+y3 by x2+xy − y2. 21. ao – bo by a3 – b3. 22. 29-y0 by x2+xy+y2. 23. x-2y14-7x3y1 — 7xy12+14x3ys by x-2y2. 24. a3+3a2b+b3 −1+3ab2 by a + b −1. 29. 30. 26. a12 - b12 by a2 – b2. a3-2a2x+27ax2 - 27x3 by a – 3x. 31. a2+8a5 by fa2+ac. 32. a-a3- Ja2+a+1 by a2--a. 33. 36x2+12+1-4xy-6x+y by 6x-3y34.5-213x4 by za-x. 55. The following examples in division may be easily verified; they are of great importance and should be carefully noticed. and so on; the divisor being x-y, the terms in the quotient all positive, and the index in the dividend either odd or even. II. x+y = x¤ — x3y + x1y2 — x3y3+x2y1 — xy5+yo, x+y and so on; the divisor being x+y, the terms in the quotient alternately positive and negative, and the index in the dividend always odd. x2-y2 =x-y, III. = x3 — x2y + xy2 — y3, x+y 26-ye x + y = x2 - x1y + x3y2 — x2y3+xy1-y5, and so on; the divisor being x+, the terms in the quotient alternately positive and negative, and the index in the dividend always even. IV. The expressions x2+y2, x+y1, x2+y3... (where the index is even, and the terms both positive) are never divisible by x+y or x-y All these different cases may be more concisely stated as follows: xn (1) "y" is divisible by x- y if n be any whole number. +y" is divisible by x+y if n be any odd whole number. (3) "y" is divisible by x + y if n be any even whole number. x+y" is never divisible by x+y or x-y, when ʼn is an even whole number. CHAPTER VII. REMOVAL AND INSERTION OF BRACKETS. For 56. WE frequently find it necessary to enclose within brackets part of an expression already enclosed within brackets. this purpose it is usual to employ brackets of different forms. The brackets in common use are ( ), { }, []. Sometimes a line called a "vinculum" is drawn over the symbols to be connected; thus a-b+c is used with the same meaning as ahence a-b+c=a−b-c. −(b+c), and 57. To remove brackets it is usually best to begin with the inside pair, and in dealing with each pair in succession we apply the rules already given in Arts. 21, 22. Example 1. Simplify, by removing brackets, the expression a-2b-[4a - 6b - {3a-c+ (5a - 26-3a-c+2b)}]. Removing the brackets one by one, we have a – 2b — [4a – 6b – {3a − c + (5a − 2b − 3a + c − 2b)} ]. Example 2. Simplify the expression -[-2x- {3y-(2x − 3y) + (3x − 2y)} +2x]. The expression=-[-2x- {3y-2x+3y+3x-2y} +2x] EXAMPLES VII. a. Simplify by removing brackets 1. a-(b-c)+a+(b−c)+b−(c+a). 2. a-[b+{a-(b+a)}]. 3. a-[2a-{3b-(4c-2a)}]. 4. {a-(b-c)}+{b−(c− a)} − {c− (a−b)}. 5. 2a-(5b+[3c-a]) - (5a - [b+c]). 6. ---(a-b-c)]}. 7. -[a-{b-(c-a)}]-[b-{c-(a—b)}]. 8.--(--))) − (− (− y)). 9. [--(b+c-a)}]+[-{−(c+a-b)}] 10. -5x-[3y- {2x-(2y-x)}]. 11. -(-(-a)) − (− (− (− x))). 12. 3a-[a+b-{a+b+c−(a+b+c+d)}]. 15. [5x-(11y-3x)]-[5y-(3x — 6y)]. 16. -[15x-14y — (15z+12y) – (10x – 15z)}]. 17. 8x-16y-[3x-(12y-x)-8y]+x}. 58. A coefficient placed before any bracket indicates that every term of the expression within the bracket is to be multiplied by that coefficient. NOTE. The line between the numerator and denominator of a fraction is a kind of vinculum. x-5 Thus is equivalent to § (x-5). Again, an expression of the form (x+y) is often written √x+y, the line above being regarded as a vinculum indicating the square root of the compound expression x+y taken as a whole. Thus whereas √√25+144=√169=13, √25+√144=5+12=17. |