17. 3.38 – 10x® +15x+8) ** – 234 – 6.3°+ 4x2+13++ + 6 3 3x5 – 6x4 – 18x3 + 12x2 +39x + 18/1 - 10x3 +15x + 81 - 2) - 6x4 8x3 + 1 2x2 + 24x + 10 3.34 + 4x3 6x? – 12x 5 3.75 3x4 + 463 – 6x2 – 12x – +5)3-7) – 10x3+156 + 8 + (* 3 - 3x2 – 5* - 2 I 2X *4 + 33 – 3x2 – 5x – 2 5/3 3x4 + 3x3 – 9x2 – 15x – 6 X3 + 3x2 + 3x + 1 2)33* + 4x4 – 6x? *3 + 3.x2 + 3x + 1 (1 H3 + 3x2 + 3x + 1 .. G. C. M. = x3 + 3x2 + 3x + 1. 18. Let c be contained p times in a, and 9 times in b, then a =pc, b=9c, and ma nb=mpe + nqc=(mp + ng)c; hence c is contained (mp+ng) times in ma + nb, and therefore c measures ma + nb. N.B.—Some error in the copying of this example or in the setting of the original. 20. For proof of this rule see any text-book on algebra. Here G. C. M. of 21 and 28= 7, .: G. C. M. = 7(42 - xy + y). 21. (aob - ab2)2 = abo(a - b) = ab(a2 – 62)2 =abla? – 62) (a? – 62) = ab(a - b)2 (a + b)2; .. G. C. M. = ab(a - b)?, 6(x2 - 1)=6(x - 1) (x+1), .. G. C. M=2(x - 1). 22. 4(1-3 +93) = 4(x + a) (x2 – xa +a), 6(x2 - 2ax - 30%) = 6(x + a) (x - 3a); .. G. C. M. = 2(x + a). LEAST COMMON MULTIPLE. 1. (x2 - a2)=(x + a) (x -a), (x3 - a*)=(x-a) (x2 + ax + a); :. L. C. M. = (x – a) (x+a) (x2 + ax + a2) = x4 + ax3 – mox – a4. 2. (x+2)=(x + 2), (x2 - 1) = (x - 1) (x+1), = x3 + 2x2 – X – 2. 3. 15a2 + 16ab - 1562 = (3a +56) (5a - 36), 9a2 – 25b2 = (3a + 50) (3a – 50); = (ga? – 2562) (5a – 36). 4. +3 + x2 + x + 1 = (x + 1) (x2+1), L. C. M. = (x2 - 1) (x2 + 1)=(x4 – 1). 5. 12x3 + 4x2 – 3x – 1= (3.x + 1) (4x2 - 1), . . L. C. M. = (3x + 1) (2x – 1) (4x2 – 1). 6. x+ + xy2 + y4 = (x2 + xy + y2) (x2 - xy + y2), 33 - 33 = (x - y) (x2 + xy + y2); .: L. C. M. = (x+ + x2y2 + y^) (x - y). 7. Let a and b represent the two algebraical quantities, and d their G. C. M., and let a=pd, b= qd, so that p and q have no common factor. Then the least quantity that contains p and 9 will be pq, and therefore the least quantity which contains pd and qd will be pod, which is therefore the L. C. M. required of a and b. 2. ** + xy **(x2 + y) x2 + 3x + 2 _ (x + 1) (x+2)_x+I 3. m 1 1712 + 2 2mx Im(m + 2x) (a – 1) (a” +a+I) ao tati 4. a + (a + b)ax + bx2 Q3 + a'x + abx + bx? (a + x) (a2 + bx) atx 2a Simplification. 1. }{x(x + 1) (x + 2) + x(x - 1) (x - 2)} + 3(x - 1) a(x+1) = }{x(x2 + 3x + 2) +x(x2 – 3x+2)} + (x - 1) ) 3 = 2r2 al - x2-1 a-X a2 - 19 = 0. 5. 5.5 + 62(6 + ax) bo(6 + ax) ax) 6+ ax a3 – 63 +22 b(a? +62) a2 {09+a= a(a - b) |