15. From 5 x take 3. 19. From 33 xy2 take 19 xy2. 2 ab. take the sum of 21. From the sum of 11 and 8x 10x and 4x. SUBTRACTION OF POLYNOMIALS. 63. When the subtrahend is a polynomial, each of its terms is to be subtracted from the minuend. Hence, To subtract one polynomial from another, change the sign of each term of the subtrahend, and add the result to the minuend. It will be found convenient to place the subtrahend under the minuend, similar terms being in the same vertical column. 64. 1. Subtract 5 x2y - 3ab + m2 from 3x2y - 2ab +4n. Changing the sign of each term of the subtrahend, and adding the result to the minuend, we have Note. The student should endeavor to perform mentally the operation of changing the sign of each term in the subtrahend, as shown in the following example : 2. From 5 a3 — 763 – 2a2b subtract 3 a2b-4ab2-2b+a3 6. From a2 + 2ab + b2 take a2 – 2ab + b2. 7. From 7 abc-11x+5y-48 take 11 abc+3x+7y+100. 8. Subtract 3m + y2 - 5a - 7 from 5m - 3y2 + 7а — 6. 9. Subtract 17 x2 + 5y2-4 ab + 7 from 31 x2-3y2 + ab. 10. From 6a+3b-5c+1 subtract 6a - 36 - 5с. 11. From 3m -5n+r - 2s take 27 + 3n - m - 58. 12. Take 4 a-b+2c- 5d from d - 3 b + a - с. 13. From m2 + 3 n3 subtract - 4m2 - 6 n3 + 71x. 15. From a - b - c take the sum of -2a+b+c and a - b + c. 16. From x2 + 2 x − 3 x + 4 take 3x + 3x2 + 5x - 7. 17. From 4 a3-3ab2 - 563 subtract 6a2b - ab2 + 46. 18. From a2 - 8 + 2a2 - 3a take 6a 11 5a2-2a. 19. Take 2x2 - y2 from the sum of x2-2xy + 3y2 and xy - 4 y2. 20. From the sum of x + 2y - 3z and 3y - 4x+z take z-5x+5y. 21. From 7 a2+3-5a2 + a −5a2 subtract 2a- 6a2 - 2 α3 + 9 — 11 a*. 22. From -7y3 + 3x2y - 2x2 + 6 xy2 subtract 8x2y - 2 xy2 + x3 - 9y. 23. From the sum of 2x - x + 5 and x2 + 8x - 11 take the sum of x-9x2 11x and - 4x + 3x2 - 6. 24. From the sum of a2+ab+b2 and a2-4ab+562 take the sum of 4 a2 + 7b2 – 2ab and 3 ab – а2 — 262. 25. From 3x2 - 7 y - 2 + xy - 5 у2 subtract - 5 xу + 6x - 2x2 - 8 + 2y2. 26. From 3x - 8x + 3x3 - 5x2 - 2x subtract - 3x2 + 4x + 6x2-6x+2. 27. From the sum of 2x - x2y - 5 xy2 and 3x2y - 5 ху2 - 4 y take the sum of -2x2-7x2y-6y and 6 ху2+5y. 28. From the sum of a 1 and 2 a3- 10 a2 — 7 a subtract the sum of - 3a2 + 2a2 - 5a and 5 α3 - 12a2+3. Note. In Arithmetic, addition always implies augmentation, and subtraction diminution. In Algebra this is not always the case; for example, in adding 2 to 5, the sum is 3, which is less than 5. Again, in subtracting 2 from 5, the remainder is 7, which is greater than 5. Thus the terms Addition, Subtraction, Sum, and Remainder have a much more general signification in Algebra than in Arithmetic. IV. USE OF PARENTHESES. 65. The use of parentheses (Art. 20) is very frequent in Algebra, and it is necessary to have rules for their removal or introduction. 66. The expression 2a-3b+(5b-c+2d) indicates that the quantity 5b-c+2d is to be added to 2a-3b. If the addition be performed, we obtain (Art. 55) 2a-3b+5b-c+2d. Again, the expression 2a-3b-(5b-c+2d) indicates that the quantity 5b-c+2d is to be subtracted from 2a-3b. If the subtraction be performed, we obtain (Art. 63) 2a-3b-5b+c-2d. 67. It will be observed that in the first case the signs of the terms within the parenthesis are unchanged when the parenthesis is removed; while in the second case the sign of each term within is changed, from + to-, or from to +. We have then the following rule for removing a parenthesis: A parenthesis preceded by a + sign may be removed without altering the signs of the enclosed terms. A parenthesis preceded by a sign may be removed, if the sign of each enclosed term be changed, from + to -, or from to +. 68. Since the brackets, the braces, and the vinculum (Art. 20) have the same signification as the parenthesis, the rule for their removal is the same. It should be observed in the case of the vinculum that the sign apparently prefixed to the first term underneath, is in reality the sign of the vinculum. Thus, + a -band-a-b are equivalent to + (a - b) and - (a - b), respectively. EXAMPLES. 69. 1. Remove the parentheses from 2a-36- (5а - 46) + (4a – b). By the rule of Art. 67, the expression becomes 2a-36-5a+4b+4a-b=a, Ans. In this Parentheses are often found enclosing others. case they may be removed in succession by the rule of Art. 67, and it is better to remove first the innermost pair. 2. Simplify the expression 2x-x 4x-3x + (- 2 х – х – а) } . We remove the vinculum first, and the others in succession. Thus, 4x - {3x + (-2x – x – a) } = 4 x {3x + (-2x-x + a)} =4x-3x + 2x + x - a = 4x-a, Ans. Reduce the following expressions to their simplest forms by removing the parentheses, etc., and uniting similar terms : 6. m2 - 2n + {a-n+3m2}-5a + 3n - m2. 7. a2-b2 - (a2 – 2ab + b2) – [a2+2ab+b2]. 8.3а-(2a-a +2}). |