case we find what the divisor must be multiplied by in order to agree with the dividend so far as certain terms which contain a are concerned, and in the second we find what the divisor must be multiplied by in order to agree with the dividend so far as certain terms which contain b are concerned. When therefore we have to divide one expression by another, both expressions being arranged in the same way, it must be understood that this arrangement is to be adhered to. 74. Def. A relation of equality which is true for all values of the letters it contains, is called an identity. The following identities can easily be verified, and should be remembered: (x2 + 2αx+a3) ÷ (x + α) = x+a. (x2 - 2αx + a3) ÷ (x − a) = x − ɑ. (x3 ‡ a3) ÷ (x − a) = x2 ± ax + a2. (x1 — a1) ÷ (x − a) = x3 ± ax2 + a2x ± a3. (x2 + a2x2 + a*) ÷ (x2 + ax + a2) = x2 ± ax+ a2. (x3+ y3 + z3— 3xyz) ÷ (x + y + z) = x2 + y2 + z2 — yz — zx — xy. m6- 6mn5 + 5no by m3 – 2mn + n3. Divide 1-7x+6x2 by (1−x)3. 8. Divide 9. 10. Divide 3 +x*y + x3y2 + x2y3 + xy* + y5 by x2 + xy + y2. x — 5x1y +7x3y2 — x2y3 — 4xy1 + 2y by x3-3x2y + 3xy3 — y3. 17. Divide a2 - 2b2 - 6c2 +ab-ac +7bc by a- b+2c. 18. Divide a2 + 2b2 − 3c2 + bc + 2ac + 3ab by a+b−c. 6a1 + 4b1 – a3b + 13ab3 + 2a2b2 by 2a2 + 4b2 3ab. 4 Divide x + y − z* + 2x3y2 + 2x2-1 by x2+ y2 — x2 + 1. 21. Divide a3-3a2b+3ab2 — b3 — c3 by a 22. Divide a3 + 8b3 − c3 + 6abc by a + 2b 23. Divide a3 + 8b3 + 27c3 – 18abc by a2 + 4b2 + 9c2 – 6bc - 3ca - 2ab. 24. Divide 27a3 - 8b3 — 27c3 – 54abc by 3a-2b-3c. 25. Divide acc3 + (ad − bc) x2 — (ac + bd) x + bc by ax – b. 2a2x2 - 2 (b−c) (3b — 4c) y2 + abxy by ax + 2 (b−c) y. 27. Divide 9a2b3 – 12a1b + 3b5 + 2a3b2 + 4a5 - 11ab1 by 3b3 + 4a3 – 2ab2. 28. Divide x3 + y3 by x + y; and from the result write down the quotient of (x + y) +23 by x+y+z. 3 29. Divide a3 - y3 by x y; and hence write down the quotient of (x + y)3 - 823 by x + y − 2z. CHAPTER VI. FACTORS. 75. Definitions. An algebraical expression which does not contain any letter in the denominator of any term is said to be an integral expression: thus a3b - 163 is an integral expression. An expression is said to be integral with respect to any particular letter, when that letter does not occur in the x2 denominator of any term: thus + is integral with α a+b respect to x. An expression is said to be rational when none of its terms contain square or other roots. 76. In the present chapter we shall shew how factors of algebraical expressions can be found in certain simple cases. We shall only consider rational and integral expressions; and by the factors of an expression will be meant the rational and integral expressions which exactly divide it. 77. Monomial Factors. When some letter is common to all the terms of an expression, each term, and therefore the whole expression, is divisible by that letter. Such monomial factors, if there be any, are obvious on inspection. 78. 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 at once. Thus, from the known identity we have a2-4b2=a2 - (2b)2 = (a+2b) (a – 2b), and we have a3+b3 = (a + b) (a2 — ab+b2), a3 +8b3=a3 + (2b)3 = (a+2b) { a2 − a (2b) + (2b)2} =(a+2b) (a2 - 2ab+4b2), 8a3 +27b6 = (2a)3 + (3b2)3 = (2a+3b2) {(2a)2 — (2a) (362) + (362)2} · (2a+3b2) (4a2 – 6ab2+9b4), The following are additional examples of the same principle: (i) (a+b)-(c+d)2 = {(a + b) + (c + d)} {(a+b) − (c+d)} =(a+b+c+d) (a + b − c − d). (ii) 4a2b2 – (a2 + b2 − c2)2 = {2ab+ (a2 + b2 − c2) } { 2ab − (a2 + b2 − c2)} ; and, since and 2ab+a2+b2 - c2 = (a + b)2 − c2 = (a+b+c) (a + b −c), we have finally 4a2b2 - (a2+b2 — c2)2 = (a+b+c) (b+c− a) (c + a−b) (a+b−c). (iii) (a+2b)3 – (2a+b)3 = {(a +2b) − (2a +b)} {(a+2b)2 + (a+2b) (2a+b) + (2a + b)2} =(ba) (7a2+13ab +762). 79. Factors of x2+ px + q found by inspection. From the identity (x + α) (x + b) = x2 + (a + b) x + ab, it follows conversely that expressions of the form x2 + px + q can sometimes, if not always, be expressed as the product of two factors of the form x + α, x+b. We shall presently give a method by which two factors of x2+px + q of the form x+a and x+b can always be found; but whenever a and b are rational, the factors can be more easily found by inspection. For, if (x+a) (x +b), that is x2+(a + b) x + ab, is the same as x2 + px+q, we must have a + b and ab= q. Hence a and b are such that their sum is p, and their product is q. = p For example, to find the factors of x2+7x+12. The factors will be x+a and x+b, where a+b=7 and ab=12. Hence we must find two numbers whose product is 12 and whose sum is 7: pairs of numbers whose product is 12 are 12 and 1, 6 and 2, and 4 and 3; and the sum of the last pair is 7. Hence x2+7x+12= (x+4) (x+3). Again, to find the factors of x2 − 7x+10. We have to find two numbers whose product is 10, and whose sum is 7. Since the product is +10, the two numbers are both positive or both negative; and since the sum is -7, they must both be negative. The pairs of negative numbers whose product is 10 are 10 and -1, and -5 and 2; and the sum of the last pair is -7. Hence x2- 7x+10= (x-5) (x-2). We have to find two Again, to find the factors of x2+3x-18. numbers whose product is 18 and whose sum is 3. The pairs of numbers whose product is - 18 are - 18 and 1, -9 and 2, -6 and 3, -3 and 6, -2 and 9 and 1 and 18; and the sum of 6 and -3 is 3. Hence x2+3x-18=(x+6) (x-3). It should be noticed that if the factors of x2 + px + q be x+a and x+b, the factors of x2 + pxy+qy will be x+ay and x+by; also the factors of (x + y)2 + p (x + y) z + qz2 will be x + y + az and x + y + bz. |