and 3. Show that n3 = n(n − 1) (n − 2) + 3 n(n − 1) + n, n1 = n ( n − 1 ) ( n − 2) (n−3)+6 n(n − 1) (n−2)+7 n(n−1)+n. 4. Find the algebraical expression which, when divided by x2 - 2x + 1, gives a quotient x2 + 2 x + 1 and a remainder x + 1. 5. Solve (i.) 3(x+3)2 + 5(x + 5)2 = 8(x + 8)2. 2. Find the sum, the difference, and the product of a + b and ab - 3(a− 1)(b − 1) + 3(a − 2) (b − 2) − (a − 3) (b − 3) = 0, and that ab - 4(a - 1)(b − 1) + 6(a − 2) (b − 2) — 4(a − 3) (b − 3) + ( a − 4) (b − 4) = 0. 4. Divide a3+b3+c3-3 abc by a+b+c, and from the result write down, without division, the quotient when 8 x3 + 8 y3 + 23 is divided by 2 x + 2y + z. 5. Solve the equations: 12 xyz (i.) x = đ (5+2), y = }(−1). (ii.) ax + by = a3, bx + ay = b3. 6. A and B each shoot 30 arrows at a target. B makes twice as many hits as A, and A makes three times as many misses as B. Find the number of hits and misses of each. D. 1. Find the value of abc(ab+be+ cd + da) ÷ (a + b) (a + c) (a + d), when a = 1, b = 3, c = − 5, and d = 0. 2. Simplify (a + b)2 – [ 2 a2 — { ( a − b ) (a + 2 b) — b(a — b)}]. 3. Find the continued product of x + a, x + b, and x + c; and from the result write down the continued product of a x, α-y, and a-z. 4. Divide 75 a2b3c5 — 15 a3b1c3 by 5 a2bc2, and 6. A square grass plot would contain 69 square feet less if each side were one foot shorter. How many square feet does it contain? E. 1. Simplify b2 + {b(a − c) + ac} − a {b − a − c } + b ( c − b − a) — ab. 2. From what must the sum of 3 a2+2 ab· ac, 3b2+2 bo - ab, and 3 c2+2 ca be be taken in order that the remainder may be a2 + b2 + c2? 3. Show that (x + y)2 + (y + z)2 + (≈ + x)2 +2(x + y) (x + z) + 2(y + z) (y + x) + 2(z + x) (z + y) · 4 (x + y + z)2. 4. Divide + 11 x2 − 7 x + by x + 2, and acx3 + (ad − bc)x2 − (ac + bd)x + bc by ax − b. 6. Find two numbers such that twice their difference is greater by unity than the smaller number, and is less by two than the larger number. F. 1. Find the value of 3(x + y + z)3 − (x3 + y3 + z3), when x=3, y=-5, and z = 7. 2. Find the continued product of a1 + x1, a2 + x2, a + x, and a-x. 3. Show that (1 − a)(1 − b) + a(1 -- b) + b = 1, and that (1 − a) (1 − b) (1 − c) + a(1 − b) (1 − c) + b(1 − c) + c = 1. 4. Divide 4 y9y2+6y1 by 2 y2 + 3y - 1. 5. Solve the equations: (i.) (a + x) (b + x) = (c + x) (d+x). (ii.) 3x7y = 4, 7x - 9 y = 2. 6. If 12 lbs. of tea and 3 lbs. of coffee cost $8.76, and 12 lbs. of coffee and 3 lbs. of tea cost $6.24, what is the price per lb. of tea and of coffee? 110. Definitions. CHAPTER X. FACTORS. An algebraical expression which does not contain any letter in the denominator of any term is called an integral expression. Thus a + b and 1⁄2 a2 – 1 b2 are integral expressions. An expression is said to be integral with respect to any particular letter when that letter does not occur in the denominator of any term. x2 x Thus + а a + b is integral with respect to x. An expression is said to be rational when none of its terms contain square or other roots. 111. In the present chapter we shall show how factors of rational and integral algebraical expressions can be found in certain simple cases. In arithmetic we mean by the factors of a number its integral divisors only; and similarly, by the factors of an algebraical expression, we mean the rational and integral expressions which exactly divide it. 112. Monomial Factors. When any letter, or group of letters, is common to all the terms of an expression, each term, and therefore the whole expression, is divisible by that letter, or group of letters. Monomial factors, if there are any, can be seen by inspection, and the whole expression can be at once written as the product of the monomial factor and its co-factor, as in the above examples. In what follows therefore we need only consider expressions which have been freed from monomial factors. 113. Factors found by comparing with Known Identities. Sometimes an algebraical expression is of the same form as some known result of multiplication: in this case its factors can be written down. We proceed to apply this principle in the case of the most important forms of algebraical expressions. 114. We know that and a2+2ab+b2 = (a + b)2, a2-2ab+b2 (a - b)2. = |