54. CHAPTER VII. REMOVAL AND INSERTION OF BRACKETS. Quantities are sometimes enclosed within brackets to indicate that they must all be operated upon in the same way. Thus in the expression 2a-3b-(4a-2b) the brackets indicate that the expression 4a-2b treated as a whole has to be subtracted from 2a-3b. It will be convenient here to quote the rules for removing brackets which have already been given in Arts. 24 and 25. When an expression within brackets is preceded by the sign +, the brackets can be removed without making any change in the expression. When an expression within brackets is preceded by the sign the brackets may be removed if the sign of every term within the brackets be changed. Example. Simplify, by removing brackets, the expression (2a-3b) - (3a + 4b) − (b − 2a). The expression = 2a-3b-3a-4b-b+2a = a-8b, by collecting like terms. 55. Sometimes it is convenient to enclose within brackets part of an expression already enclosed within brackets. For this purpose it is usual to employ brackets of different forms. The brackets in common use are ( ), { }, [ ]. 56. When there are two or more pairs of brackets to be removed, it is generally best to begin with the innermost pair. In dealing with each pair in succession we apply the rules quoted above. Example. Simplify, by removing brackets, the expression a-2b-[4a - 6b - {3a - c + (2a-4b+c)}]. Removing the brackets one by one, the expression = a - 2b - [4a - 6b - {3a-c+2a-4b+c}] = a -2b-4a+6b+3a - c+2a-4b+c = 2a, by collecting like terms. Note. At first the beginner will find it best not to collect terms until all the brackets have been removed. 10. m-(n-p) - (2m - 2p+ 3n) − (n − m +2p). 11. a-b+c-(a+c− b ) − (a+b+c) − (b + c − a). 12. 5x-(7y+3x) − (2y+7x) − (3x+8y). 18. p2-242-(y2+2p2) - {p2+3g2 - (2p2 - g2)}. 19. x-[y+(x-(y-x)}] 20. (a-b)-{a-b-(a+b)-(a - b)}. 21. p-[p-(q+p) - {p− (2p − q)}]. 22. 3x-y-[x-(2y-z)-(2x-(y-z)}] 23. 3a2 -[6a2 - {8b2 – (9c2 – 2a2)}]. 24. [3a-{2a- (a - b)}]-[4a - {3a - (2a - b)}]. 57. A coefficient placed before any bracket indicates that every term of the expression within the bracket is to be multiplied by that coefficient; but when there are two or more brackets to be considered, a prefixed coefficient must be used as a multiplier only when its own bracket is being removed. Examples 1. 2x + 3(x − 4) = 2x + 3x − 12 = 5x − 12. 2. 7x-2(x-4)=7x-2x+8=5x+8. Example 3. Simplify 5a-4[10a +3{x − a − 2(a+x)}]. =5a-4[10a +3{x - a -2a - 2x}] =5a-4a+12x = a + 12x. On removing the innermost bracket each term is multiplied by -2. Then before multiplying by 3, the expression within its bracket is simplified. The other steps will be easily seen. 58. 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 a (b+c), and hence a-b+c=a-b-c. NOTE. The line between the numerator and denominator of a x-5 fraction is a kind of vinculum. Thus is equivalent to 3(x-5). Example 4. Find the value of 3 84-7-11x-4{ - 17x+3(8 - 9 - 5x)}]. The expression = 84-7[-11x-4{ - 17x+3(8-9+5x)}] = 84-7[-11x+8x+12] = 84+21x - 84 = 21x. When the beginner has had a little practice the number of steps may be considerably diminished. Insertion of Brackets. 59. The rules for insertion of brackets are the converse of those given on page 12, and may be easily deduced from them. For the following equivalents have been established in Arts. 24 and 25: a+b-c=a+(b −c), a-b-c=a-(b+c), a−b+c=a− (b −c). From these results the rules follow. Rule. 1. Any part of an expression may be enclosed within brackets and the sign + prefixed, the sign of every term within the brackets remaining unaltered. Examples. a-b+c-d-ea−b+ (c-d-e). x2 - ax + bx - ab = (x2 − ax) + (bx – ab). Rule. 2. Any part of an expression may be enclosed within brackets and the sign - prefixed, provided the sign of every term within the brackets be changed. Examples. a-b+c-d-e a − (b −c) - (d+e). xy-ax − by + ab = (xy – by) − (ax – ab). 60. The terms of an expression can be bracketed in various ways. Example. The expression ax - bx + cx − ay + by − cy may be written or or (ax − bx) + (cx − ay) + (by − cy), (ax-bx + cx) - (ay −by+cy), (ax-ay) - (bx - by) + (cx − cy). 61. When every term of an expression is divisible by a common factor, the expression may be simplified by dividing each term by this factor, and enclosing the quotient within brackets, the common factor being placed outside as a coefficient. 5. 8(x-3)- (6 − 2x) − 2(x+2)+5(5−x). 6. 2x-5(3x-7+y)+4(2x+3y-8) -7y. 7. 2x-5{3x-7(4x – 9)}. 8. x3+3(x2y+xy2) + y3 − x3-3(x2y — xy3) — y3. 9. 4x-3x-(1-y)+2(1-x)}. 10. x-(y-z)-[x-y-z-2(y+2)]. 11. a2-[x2-{x2 - (z2 - x2 - y2) - 2y2}+y3]. 12. 5x+4(y- 2z) — 4{x + 2( y − z)}. 13. a+ {-2b+3(c-d-e)}. 14. {a-(b2-c2)}- [2a2 - {a2 - (b2 - c2)} – 2(b2 - c2)]. 15. 3p-{5q-[6g+2(10q − p)]}. 16. 3x-2[2x-(2(x-y)-y}-y]. 18. 12-[6a-(7 − a − 5) - {5a + (3 −2 − a)}]. 19. b2-{a2+ab-(a2+b2)} - [a2 - {3ab - (b2-a2)}]. 20. 2[4x-{2y + (2x − y) − (x + y)}] − 2( − x − y −- x). 21. 20(2-x)+3(x-7) - 2[x+9-3{9-4(2-x)}]. 22. -4(a+y)+24(b − x) −2[x+y+a−3{y+a− 4(b+x)}]. 23. Multiply 2x-3y-4(x-2y) +5{3x − 2(x − y)} by 4x (y-x)-3(2y-3(x+y)}. In each of the following expressions bracket the powers of x so that the signs before the brackets may be (1) positive, (2) negative. 24. ax1+2x3- cx2 + 2x2 - bx3 – x1. 25. ax2+a2x3- bx2 -- 5x2- cx3. MISCELLANEOUS EXAMPLES II. 1. Find the sum of a -2b+c, 3b - (a−c), 3a - b+3c. 3. Simplify a +2b − 3c + (b − 3a + 2c) – (3b − 2a – 2c). 4. Find the continued product of 3x2y, 2xy2, -7xy3, -5x1y3. 5. What quantity must be added to p+q to make 2q? And what must be added to p2 - 3pq to make p2+2pq+q2? 6. Divide 1-6x+5x3 by 1-x+3x2. 7. Multiply 3b2+2a2 -5ab by 2a+3b. 8. When x=2, find the value of 1 − x + x2 2.3 9. Find the algebraic sum of 3ax, - 2xz, 9ax, −7xz, 4ax, - 4xz. 10. Simplify 9a − (2b − c) + 2d − (5a + 3b) + 4c − 2d, and find its value when a = 7, b=-3, c = -4. - 11. Subtract ax2-4 from nothing, and add the difference to the sum of 2x3- 5x and unity. 12. Multiply 3x2y − 4xy3z+2x3y2z3 by -6x2y2z3 and divide the result by 3xyz2. 13. Simplify by removing brackets 5[x-4{x − 3(2x − 3x + 2) }]. 14. Simplify 2x2 - (2xy - 3y2)+4y2 + (5xy − 2x2) + x2 − (2xy +6y2). 15. Find the product of 2x-7y and 3x+8y, and multiply the result by x+2y. 16. Find the sum of 3a+2b, -5c-2d, 3e +5ƒ, b-a+2d, -2a-3b+5c-2f. 17. Divide 4 – 4x3 – 18x2 - 11x+2 by x2-7x+1. 18. If a = -1, b=2, c=0, d = 1, find the value of adac-a2 - cd + c2 − a +2c+ a2b+2a3. |