19. Subtract 3x2 from the sum of 2 and - 4 x. 7 xm from → 2 yr. - ab, ac, ad. 20. Add - 5 x2 to the result of subtracting - 2 x from 0. Simplify the following expressions by uniting like terms: 21. a + 1+ a − 1. 22. 2x + 5+ 3 x −7.. 24. 3x2-4 y2 + 2 x2 — 6 y2. 5c-8 a 25. a+b-3 a + c 4b+ 6 a 28. 3 (a + m) 3 c + 11 b. m1c + 11. a4b+ 3 bx - 2 ab1 + 10 a1b — 2 bx — aba. · 4 (a + m) − 2 (a + m) + 8 (a + m). 29. (a + z)8-2 (a + z)3 + 2 (a + 23) + 7 (a + z)3 − 5 (a + z3). Simplify the following expressions, first removing parentheses: 31. 5x-(-2y+3x). 35. 2xy +5yz- (2xy - 3 yz)-[2xy - (3xy - 2yz)+5yz]. Find the values of the expressions in Exx. 30-35, 36. When a = 1, x = 3, y = 5, z = 10, m = 4, n = — 7. 37. When a = — 3, x = 6, y = 7, z = 8, m = — Simplify 38. a+ (a+1)+(a + 2) + (a + 3). 41. Find the sum of 7 terms, the first term being x2, and each succeed ing term being 1 less than the preceding term. 42. Find the sum of 6 terms, the first term being m+n, and each succeeding term being p less than the preceding term. Addition and Subtraction of Multinomials. 4. Ex. 1. Add -2a+3b to 3 a - 5 b. We have (3a-5b) + ( − 2a+3b) = 3a5b-2a+3b, =a-2b. Ex. 2. Subtract - 2a+3b from 3a5b. We have (3a5b) − (−2a+3b)=3a-5b+2a-3b, =5a-8b. In adding multinomials, it is often convenient to write one underneath the other, placing like terms in the same column. Ex. 3. Find the sum of -4x+3y2 - 8 z2, 2x2 - 3x2, and 2 y2+5 x2. It is evidently immaterial whether the addition is performed from left to right, or from right to left, since there is no carrying as in arithmetical addition. Changing mentally the signs of the terms of the subtrahend, and adding, we have When several multinomials are to be subtracted in succession, the work is simplified by writing them with the signs of the terms already changed. We then have 15. x3 — 3x2y + 3 xy2 — y3 from x3 + 3x2y + 3 xy2+ y3. 16. 2x4 - 3 x3 − 7 x2 + 3 x + 1 from 2 + 4 x − 6 x2 - 2 x3 + 3x4. 20. 7a9b (x + y)2-7(x + y) +3. c, 5a-3b-2c, 2a+3b-5 c. 21. 3x2-5x+1, 7x2 + 2 x − 3, x2 - 2x 3. ax + a2, 2x2 + 3 ax - 4 a2, x2 + ax + 2 a2. - 4 ab + b2, a2 — 2 ab — 2 b2, 2 a2 - 3 ab + 4 62. 24. a2 - 2 ab + 2 b2, 2 a3 – 3 ab + b3, a2 + 5 ab — b3. 25. 2x2y3 + 4x3y2, -5x2y3 + 2x2y2 - 3 x3y2, 4 x3y2 - 5 x2y3 — 6 x2y2. — - 2a+5b3c, - 2a + b 3 x2 + 4 x − 2, 5 x3 + 4 x2 + 5, a3 — a2, a2 a+1, a3 - 2 a2 — a 2. 29. 2 a + b · (c + d), a + (b − c) — d, a + b − (c − d). a + b − (2 c + d). 31. 3(a+b) 4(a + b)2 + 5(a + b)3, -(a+b)8+2(a + b) 2 − (a + b). 32. (a + b)2 — 2(a + b)3, C. 7(x2+y2)-3(x2-y2)+2xy, 2(x2-y2)-4 xy, 3(x2+y2)-(x2-y2). 33. Subtract the sum of a2 + ab + b2 and ab from 2 a2 + 3 ab + 2 b2. 34. Subtract the sum of a2 and b2 + c2 from the sum of b2 and a2 - c2 37. What expression must be added to 2a-3b+4c to give 4 a + 26-2c? 38. What expression must be added to xy+xz+yz to give x2+y2+z2 ? 39. What expression must be subtracted from a2 + ab + b2 to give a2 - 2 ab + b2 ? 40. What expression must be subtracted from x2 - 2xy + y2 to give x2 + 2xy + y2? 41. What expression must be added to x2 + x + 1 to give 0 ? 2a-3b+4c, y=-3a+2b-7c, z=9a-7b+6c, find If x = the values of 43. xy + 2. 44. x+y2. 45. xy- 2. 42. x + y + 2. x = 5a2 - 3 ab + b2 - 3 ac + 2 bc + c2, z = 4a2 7 ab+5b2-4 ac· 5 bc + c2, This example illustrates the following principle: The product of two or more powers of one and the same base is equal to a power of that base whose exponent is the sum of the exponents of the given powers; or, stated symbolically, EXERCISES V. Express each of the following products as a single power: Ex. (a1)3 = a*a*a*a*a* = a1+1+1+*+* = a1×5 = a2o. This example illustrates the following principle: A power of a power of a given base is equal to a power of that base whose exponent is the product of the given exponents; or, stated symbolically, mnp (am)n = amn; [(aTM)"]o = am2o; etc. For, (am)n = amamam. ... to n factors = am+m+m+ to n summands = amn Likewise, [(am)n]r = (amn)r = amp; and so on. EXERCISES VI. Find the values of the following powers: 1. (32)8. 2. 328. 3. (48)2. 4. [(-2)3]4. 5. (-28)5. |