3. Divide 8 a2 — 12 a5 + 8 a4 + 18 as — 30 by 6 — 4 a\ 4. Combine - y-p^ - in a single fraction, and reduce it to its lowest terms. 5. Divide x* — 2 by a; — -. ar x 6. Find (a — 6)4 and 7. Solve the equation — ——r, in which a a a -\- o a — o and b denote known quantities. Find also the value of x when a —— 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 f. 9. Solve the equations 2x — y = 5, 3 y — 2z = — 13, 2z — 4* == 2. 10. Verify the answers of Nos. 7, 8, and 9, by showing that they satisfy the original conditions. plus twenty-five gallons was wine, and one third part minus five gallons was cider; how many gallons were there of each? 10. Solve the equations | + 7 y = 99, | -f- 7 x = 51. VIII. 1. Eeduce the following expression to its simplest form: (a + 2. Multiply 2a? — 3xy + 6«/2 by 3 a? + 3 xy + 5f. 3. Divide 40 a + 8 a4 — 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: xTM x of = what? 5. Reduce the following expression to a single fraction, having the least possible denominator: ^ _ ^g — 1 1 — x (1 + xf „ 25a26«a*, 10 a 6" 6. Divide ,„ „ „ by — jr= «• 7. Find by the Binomial Theorem the first four terms of (a-^andof(l-l^)20. 8. Find the value of x in the equation x — a = -7 4" 3—> iQ which a, &, c, d, and e denote known quantity de ties. Find, also, what the value of x becomes when a = — 3, & = 0, c = — 2, d = — 2, e = 4; and verify it for this case by showing that it satisfies the equation. 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. Eeduce the following expression to its simplest form: ab — c (x — b) — [(a; -f- c) (x — c) — c (b — {c — x}) — x2]. 2. Into what two factors can the following expressions be severally resolved: (4 a;6?/2 — 25 a:16); (m3 — n3). 3. Multiply 6a3 — 2a?b + 4a?*2 by 2a*b — Sab* — 3b3. 4. Divided— 6x*— 45 a; + 3 54 by 3 a-f 3 a3 — 9. 5. State the rule for multiplying different powers of the same quantity, and give its reason. Examples: 6. Reduce to one fraction (with least possible denomina..3.2.2 1 4 a8 Ifi 2cbs 7. Divide ^7-^ by — t „0; and raise the quotient to the second power. 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? 9. Solve the equations \x-\-\y = \z — 1, 2z — \y = 28-fz,i^-2 = y + 5. 10. Solve the equation x2— 5a;— 6 = 0; and verify the answers by showing that they satisfy the equation. 11. Show that no binomial can be an exact second power. |