51. When dissimilar terms have a common literal part, this may be taken as the unit of addition. The sum of the terms will then be expressed by inclosing the sum of the coefficients in a parenthesis, and prefixing it to the common unit. Ans. (4a + 4) y + (3c + 6) x. 8. Add 3x + 2xy, bx + cxy, and (a + b) x + 2cdxy. Ans. (a + 2b + 3) x + (2cd + c + 2) xy. 11. Add (a + 2b) m — c√m, (2a—6c) m — 3a√m, (5c—4a) m -b√m, and (2a-3b) m + 4a√m. Ans. (a-b-c) (m+ √m). 12. Add ax + y + z, x + ay + z, and x + y + az. Ans. (a + 2) (x + y + z). SUBTRACTION. 52. Subtraction is the process of finding the difference between two quantities. 53. It is evident that 5 units of any kind or quality subtracted from 8 units of the same kind or quality, must leave 3 units of the same kind or quality. That is, Also, + 8a − (+ 5a) = + 3a. За. -8a-(-5a)=-3a. But these remainders are the same as we shall obtain by changing the signs of the subtrahends and then adding the results, algebraically, to the minuends. Thus, +8a(+5a) = + 8a — 5a = + 3a Hence, in Algebra, Subtracting any quantity is equivalent to adding the same quantity with its sign changed. 54. This principle may be established in a more general manner as follows: Let it be required to subtract the quantity be from a. we therefore add c to the first result, and obtain the true remainder, a−b+c. But this result is the same as would be obtained by adding −b+c to a. 55. It follows from the principle enunciated above, that any quantity is subtracted from nothing or zero, by simply changing its sign or signs. Thus, 56. From these principles and illustrations we deduce the following RULE.-I. Write the subtrahend underneath the minuend, placing the similar terms together in the same column. II. Conceive the signs of the subtrahend to be changed, unite the similar terms as in addition, and bring down all the remaining terms with their proper signs. 11. From a 5c+2 subtract 12. From 8x2 c2 · b + c2 + c. - a + c + 2. Ans. 8a-6c. · 3xy + 2y2 + c subtract x2 6xy + 3y2 — 2c. Ans. 7x2+3xy-y2+3c. 13. From a + b subtract a - b. 14. From subtract -y. + 29 Ans. 2b. Ans. y. 15. From a + b + c subtract -a-b-c. Ans. 2a+2b+2c. 3b 16. From 3a-b-2x+7 take 836 + a + 4x. 17. From 6y2-2y-5 take Ans. 2a + 2b 6x 1. 8y2-5y+12. Ans. 14y2+3y17 18. From 3p+q+r3s take q8r+ 28-8. 20. From 24-—3x3+5x2-7x+12 take x4—4x3 + 2x2-6x+15. Ans. x3 + 3x2 21. From a5-3a1c + 5a3c2-2a2c3 + 4ac4-c5 take a5. x - 3. 4a1c + Ans. ac +3a3c2 + 3a2c3 + ac1. 22. From 2x4 + 28x3 + 134x2 252x+144 take 2x4+212 + 67x2 Ans. 7x367x2 - 189x + 60. 63x+84. 23. From 25+5x+y+10x3y2+10x2y3+5xy+y3 take x5—5x1y+ 10x3y2 — 10x2y3 + 5xy1 — y3. Ans. 10x4y+ 20x2y3 + 2y3. 24. From the sum of 6xy-11ax and 8x2y+3ax3, take 4x2y 4ax3 + a. Ans. 10x y 25. From the sum of 8cdx + 15a2b 3 and 2cdx - 8a2b+24 take the sum of 12a2b - 3cdx 8 and cdx 4a2b+16. - 57. The difference of two dissimilar terms may often be conveniently expressed in a single term, as in (51), by taking some common letter or letters as the unit of subtraction. 4. From cdm+4ax2 take d2m+3ax2. Ans. (c2—1)d2m+ax2. 5. From ax + by + cz take mx + ny + pz. Ans. (a — m) x + (b − n) y. + (c — p) z. 6. From ax + bx + cx take x + ax + bx. Ans. (c-1)x. 7. From (a + 26 + c)√xy take (2b — c)√xy. Ans. (a + 2c)√xy. 8. From (3a-2m)x3 +(5a+2m)x2+(4a—m)x take (a—m)x Ans. (2am) x3 + (7a + 3m) x2 + (2a + 2m) x. 9. From 12az2+3a2z4+4a326+5a428 take 2+2a+3a2+ 4a3z. Ans. 1+(2a—1)z2 +(3a2—2a)z2+(4a3—3a2)zo + (5aa—4a3)z3. USE OF THE PARENTHESIS. 58. The term, parenthesis, will be employed hereafter as a gen eral name to designate the various signs of aggregation employed in algebraic operations. The following rules respecting the use of the parenthesis should be thoroughly considered by the learner, if he would acquire facility in algebraic transformations. 59. From the definition of the signs of aggregation (17), we understand that if the plus sign occurs before a parenthesis, all the terms enclosed are to be added, which does not require that the signs of the terms be changed; but if the minus sign occurs before a parenthesis, all the terms enclosed are to be subtracted, which requires that the signs of all the terms be changed. Hence, 1. A parenthesis preceded by the plus sign may be removed, and the enclosed terms written with their proper signs. Thus, -α- b + c d + e. a - b + (c − d + e) 2. Conversely: Any number of terms, with their proper signs, may be enclosed by a parenthesis, and the plus sign written before the whole. Thus, abc-d+e=a+(− b + c −d+e). 3. A parenthesis preceded by the minus sign may be removed, provided the signs of all the enclosed terms be changed. Thus, e) = a ⋅ b + c⋅ d + e. 4. Conversely: Any number of terms may be enclosed by a parenthesis, preceded by the minus sign, provided the sign of every term thus enclosed be changed. Thus, α b + c d + e = a 60. When two or more parentheses are used in the same expression, they may be removed successively by the above rules. Thus, a - — {b—c—(d—e) } = a—{b―c―d+e}=a−b+c+d—e. Or, in a different order, a — {b―c—(d—e) } − a−b+c+(d—e) = a−b+c+d—a. |