2d. When the signs are unlike, the difference between the positive and negative coefficients prefixed to the common literal part, with the sign of the greater, gives the result. Hence the rule: 34. To add similar terms. I. When the signs are alike, prefix the sum of the coefficients with the given sign to the common literal part. II. When the signs are unlike, find the sum of the positive and the sum of the negative coefficients, separately, and prefix the dif ference of these sums with the sign of the greater to the common literal part. 35. To add polynomials. I. Write the quantities in separate columns, placing similar terms under each other. II. Add each column separately, and connect the several results with the proper sign. 5. 18 xyz3, 13 xyz3, -21 xyz3, - 40 xyz3. 6. 1a2b, — 1a2b, — 1a2b, + 2 a2b. 7. a2bcd, 8 ab + 4 c3d, 23 ab - 16 c3d. 8. 4x+3y+2z, 3x-4y-3z, 8x-3y+4z, x-8y-z. 9. a+ab+ ab3c, 3 a +2ab+4ab2c, 8a36a2b-16 ab2c. 10. 3(a+b2), 5(a + b2), 6(a + b2), − 16(a + b2). 3 m2 3 m3 4m, 18 1 8 m3 - 8 m2 + 14 m, 13 m - 8 m2 - 13 m3. Sa2 + b + c − d, a2 + b — c, a2 — d, 12. S b-c+4d, 9 a2+10b-6c+ 12 d. Commutative Law. No matter what the order of the terms in addition, the result will be the same. Quantities may be grouped in any manner, and added together; this process is called the associative law of addition. Thus, a + b + x + y + z = (a + b) + (x + y + z) = a + (b + x + y) + z. 36. The Unit of Addition is the common letter or quantity whose coefficients are added in finding the sum of two or more quantities. Thus, x is the unit of addition in 13x+6x= 19 x. Again, a+b+c is the unit of addition in 3(a+b+c) + 8 (a + b + e) − 10 (a + b + c) = (a + b + c). 37. Dissimilar terms may have a unit of addition. Thus, x is the unit of addition in Add the following: 1. ax2, bcx2, dcx2, 12 x2. EXERCISE 4. 2. axy + cxy, bxy + exy, m2xy + n2xy. 3. abc + efbc + xybc, mnbc + pqbc + edbc. 4. (a−b) (p + q), (x − y) (p + q), (m − n) (p + q). 5. 2a√x+y, 6 b√x+y, − 8 c√ x + y. SUBTRACTION. 38. Subtraction is the process of finding the difference between two quantities. 39. The student must bear in mind that the terms addition and subtraction have a more general sense in Algebra than in Arithmetic. In Arithmetic all quantities are regarded as positive; consequently the sum of two quantities will be greater than either, while the difference will be less than either. In Algebra the sum of two quantities may be less than either, while the difference between two quantities may be greater than either. Using letters instead of figures for the purposes of this demonstration, we have: +a+(+2a) = 3 a + a(+2a) = — a From the above we see that subtracting a positive number is equivalent to adding an equal negative number, and subtracting a negative number is equivalent to adding an equal positive number. 41. Hence the rule: To subtract one quantity from another. I. Change the sign of each term of the subtrahend and proceed as in addition. 9. From m2+3n-4p+6 g subtract 3 p+5q-3 m2 + n. 10. From 3 (p + q) subtract 9 (p + q). 11. From 3 ax (m2 + n − p) subtract 2 ax (m2 + n − p). 12. From p+q-m subtract p+9+ m. 13. From 2 a √b+ √c subtract 5 a √b – √c. 14. From 9 xy (2 √x + 3 √y − √c subtract 8 xy (2 √x + 3 √y – √e). 15. From 25 - 23+ √x subtract a* - 2 √x + x3. x PARENTHESES. 42. From section 40 it becomes apparent that if a parenthesis is removed when a plus sign precedes it, the sign of the term within the parenthesis remains the same, also that if a parenthesis is removed when a minus sign precedes it, the sign of the term within the parenthesis is changed. The same rule holds when the parenthesis encloses several terms. 43. Hence, in removing parentheses, we have the following rule: I. If a parenthesis is removed which has a plus sign before it, the terms of the parenthesis remain unchanged. II. If a parenthesis is removed which has a minus sign before it, the sign of each term enclosed in the parenthesis must be changed. 44. Sometimes several parentheses occur, one enclosed within the other. In such cases each parenthesis must be treated according to the sign before it. The best method of procedure is to remove the innermost parenthesis first. Thus : 3 x [ − 4 y — { 3 x − (2 y + x − y) + 6 y} −6 x] =3x-[-4y-(3x-(2y+x − y) + 6 y} −6 x] =3x-[-4y- {3x-2y-x+y+6y} - 6 x] = = {3 3x [-4y-3x+2y+x-y-6y-6 = 11 x + 9 y. x] |