3. Divide 8 a2-12 a5 +8a+ 18 a3-30 by 6-4 a2. fraction, and reduce it to its lowest terms. and b denote known quantities. Find also the value of x when α= 1, b: = 3. 8. Find a certain fraction which is such that if 3 be subtracted from both numerator and denominator, the value of the fraction becomes, and that if 11 be added to both numerator and denominator, the value of the fraction becomes g. 9. Solve the equations 2xy=5, 3y-2x=-13, 22-4x = 2. 10. Verify the answers of Nos. 7, 8, and 9, by showing that they satisfy the original conditions. VI. 5abc subtract 3 ac-[3 ab — 2. Divide 28 a2 6 a3— 6 a5 4 at 96 a +264 by 3a24a+11. 7. Find (xy) and (a2 — 3b)5 by the Binomial Theō rem. 8. Find a number from which if 5 be subtracted & of the remainder will be 40. 9. Solve the equations x-6z=6-2y, 3x-5y=20, 4%= 5 x 27. 10. Verify the answers to Nos. 8, 9, by showing that they satisfy the original conditions of those problems. VII. 1. From 4 a2x-(2 abc-4bc8d) subtract 8 abc(4a2x-2d) + abc. 2. Multiply x2+ xy + y2 by x2- xy + y2. 3. Divide 3 a 8 a2b2 + 3 a2 c2 + 5b4 — (a* — b1) (a2 + 2 a b + b2) plest form by inspection. 362c2 by to its sim 8. Subtract (a-26)5 from (a +26)5. Use the Binomial Theorem. 9. In a mixture of wine and cider one half the whole plus twenty-five gallons was wine, and one third part minus five gallons was cider; how many gallons were there of each ? y 10. Solve the equations+7y=99, + 7 x = 51. VIII. 1. Reduce the following expression to its simplest form: (a + b) x − (b — c) c — [(b — x) b − (b — c) (b + c)] 2. Multiply 2x3-3xy+6y2 by 3x2+ 3 xy + 5 y2. 3. Divide 40 a + 8 aa 50 a2 8 by 5 a 2 a2 2. 4. Give the rule for multiplying different powers of the same quantity, and explain its reason. Example: xm × what? 5. Reduce the following expression to a single fraction, having the least possible denominator: 1 + x 4 x (1 x)3 1 7. Find by the Binomial Theorem the first four terms of (a 6)20 and of (1 — 3,4)20. 5 y2 8. Find the value of x in the equation x-a = bc c2 x + in which a, b, c, d, and e denote known quantide' ties. Find, also, what the value of x becomes when a = 9. A and B have together as much money as C; B and C have together 6 times as much as A; and B has $680 less than A and C together have: how much has each? Eliminate by comparison; and verify the answers by showing that they satisfy the given conditions. IX. 1. Reduce the following expression to its simplest form: a b — c (x — b) — [(x + c) (x — c) — c (b — {c — x}) x2]. 2. Into what two factors can the following expressions be severally resolved: (4x6 y2 - 25 x16); (m3 — n3). 3. Multiply 6 a3-2a2b4ab2 by 2a2b-5ab2-378. 4. Divide 9 a2-6x4 — 45 x + 3x3 +54 by 3x + 3x2 - 9. 5. State the rule for multiplying different powers of the same quantity, and give its reason. Examples: xm × x2= what? (xx)=what? 6. Reduce to one fraction (with least possible denomina 8. A and B are building a wall. A alone can build it in a days, and B alone in b days. In what time can both together build it? 28 9. Solve the equations x+y={z-1, 2z-y= 10. Solve the equation 2-5x-6=0; and verify the answers by showing that they satisfy the equation. 11. Show that no binomial can be an exact second power. X. 1. Reduce the following expression to its simplest form: x3 3 — (— xy2 + x3. | xy — x2 [— { y3 — y (xy — x2)}]. 2. Reduce the following expression to its simplest form: (a + b) b + c — [(c + d) (a + d) c (a + b − 1)(a + c) (d — b)]. 3. Multiply 15 a2+18 a b― 1462 by 4a2-2 ab- b2. 4. Divide 43 x2 y2 — 22 x3 y + 24 y+8x4-38 x y3 by 3xy-2x2-4y2. 5. From 9. State the rule for multiplying different powers of the same quantity, and give its reason. xm xxn = what? (xx)=what? (a2b)m = ? denoted by at? am+n m-n =? What is 10. What is the reason that any term may be transposed from one member of an equation to the other, provided its sign is changed? |