Ex. 3. Find the G. c. M. of 2x3-15x+14 and 4-15x2+28x−12. x-15x2+28x-12 2 2x3-15x+14) 2×1 – 30x2+56x−24 (x 24-15x2+14x -15x2+42x-24 -3(5x2-14x+8) Rejecting -3 and multiplying each term of 2x3 – 15x+14 by 5, we get 5x-14x+8) 10x3–75x+70 (2x 10x3-28x2+16x 28x2-91x+70 7(4x2-13x+10) Rejecting the 7 and multiplying 4x2-13x+10 by 5, 5x2-14x+8) 20x2−65x+50 (4 20x2-56x+32 -9x+18 -9(x-2) x-2) 5x2-14x+8 (5x-4 5x2-10x -4x+8 -4x+8 Therefore x -2 is the G. c. M. required. Ex. 4. Find the G. C. M. of x3- (a2 + b2)x+ab2 and x3 — 2ax2 + (a2+b2) x — ab2. x3 − (a2+b2)x+ab2) x3-2ax2+(a2+b2)x−ab2 (1 x3-(a2+b2)x+aba −2ax2+2(a2+b2)x — 2ab2 ax2 — (a2+b2)x+ab2) ax3−a(a2+b2)x+a2b2 (x+(a2+b2) (a2+b2) x2-a (a2+2b3)x+a2b2 a (a2+b2) x2-a2 (a2+2b2)x+a3b2 a (a2+b2) x2 - (a1+2a2b2+b1)x+ab2 (a2 +b2) x-a) ax3- (a3+b2)x+ab2 (ax-b2 ax2-a2x -b2x+ab2 -b2x+ab2 Therefore x-a is the G. c. M. required. Ex. 5. Find the G. c. M. of 3x2-(3c+d−3)x−3c-d and 2x2+(2a+b+2)x+2a+b. 3x2-(3c+d-3)x-3c-d 2 2x2+(2a+b+2)x+2a+b) 6x2-2 (3c+d-3)x-6c-2d (3 ― (6a+3b+6c+2d) x-6a-3b-6c-2d == = Rejecting the factor −(6a+3b+6c+2d), x+1) 2x2+(2a+b+2)x+2a+b (2x+(2a+b) 2x2+2x (2a+b)x+2a+b (2a+b) x+2a+b Therefore +1 is the G. c. M. required. Ex. 6. Find the G. c. M. of €2 a3 + €2x — a3 −1 and ea2+2€*a2 — 2e*+a2 −2. = (a3+1) (e2 −1) and ea2+2€*a2 — 2€*+ a2 — 2 =a2. (e2+2+1)-2(+1) =a2. (e+1)-2(+1) =(e*+1){a2. (e*+1)−2}. Hence it appears that +1 is the c. G. M. required. Ex. XXXIII. Examples for practice. Find the G. c. M. of 1. x2+x-30 and x2+11x+30; x3-a3 and x2-a2. 2. 6x2+7x-20 and 3x2-x-4; x3- xy1 and x2-- y2. 3. x3-3x2-4x and x2-7x+6. 4. x2+11x+30 and 9x3 +53x2-9x-18. 5. x3+4x2-5 and x3-3x+2. 6. x3+4x2-5x and x3-6x+5. 7. x3+2x2-3x and 2x2+5x2 −3x. 8. 14-34x+12x2 and 42a-4ax - 6ax2. 9. x+2x2+2x+1 and x3 — 2x-1. 10. 11. 3x3 + x2-5x+2 and 15x2+11x-14 a2-5ab+4b2 and a3-a2b+3ab2 - 3b3. 12. x3-3x2+7x-21 and 2x2+19x2+35. a2+b2+c2+2ab+2ac+2bc and a2-b2-c2-2bc. 18. 3a3-3a2b+ab2 - b3 and 4a2-5ab+b2. 19. 20. 21. x-ax3+a3x-a1 and x3-a3. -2x3y+2xy3-y and -2x3y+2x2y3-2xy3+y1. 20+x2-1 and 75x+15x3-3x-3. 22. x-6x+13x2-12x+4 and 25-4x+8x3 — 16x2+16x. 23. 2a4-11a2b2+1264 and 3a6-48a2b1. 24. 2x-4x4 +8x3 — 12x2 + 6x and 3x2 – 3x2-6x2+9x2-3x. -- 25. 36a6-18a5-27a4+9a3 and 27ab2-18a4b2 — 9a3b3. 26. 3a4-a2b2-264 and 10a4+15a3b-10a2b2-15ab3. 27. x+xy-xy-y3 and x-x3y-x3y2+y3. 28. x+4x-3x1—16x3 +11x2+12x-9 and 6x+20x1 — 12x3-48x2+22m+12. 29. 2x3-16x+6 and 5x+15x+5x+15. 30. a-ba-ba1+ b5 and a3 + b2a® + ba2 +-b3. 31. 4x3y-4x3y3 + 4x2y1 — xy3 and 24x+16x1y-36x3y2-12x2y3+8xy1. 32. x2+(ab)x−ab and x2 + (a+b) x+ab. 33. x-(a2+b2) x2 + a2b2 and xa1— (a+b)2x2+2ab (a + b) x − a2 b3. and a-(a-b) x3 + (a−b) b3 x − b1. 36. ax2+(a+b) x3 + (a+b+c) x2 + (a+b+c+d) x2 + (b+c+d) x2 37. np3q+3np3q2 — 2npq3 — 2nq1 and 2mp2q2-4mp1-— mp3q+3mpq3. 38. 1+x2+x+x and 2x+2x2+3x2+3x2 39. 2x-4+2x-2+12x-1+8 and 3x-4-3x-3+21x1+15. 74. To find the Greatest Common Measure of three algebraical expressions. Let A, B, and C be the expressions; find D, the Greatest Common Measure of A and B, by Art. (73): then the Greatest Common Measure of D and C will be the Greatest Common Measure of A, B, and C. For every measure of A and B measures D, Art. (73), therefore every measure of A, B, and C measures C and D, and every measure of D is a measure of A and B, therefore every measure of D and C is a measure of A, B, and C. Hence the measures of A, B, and C are the same as those of D and C, and therefore the highest measure of C and D will be the highest measure of A, B, and C. Hence we derive the following Rule. RULE. "Find the Greatest Common Measure of any two of them, and then the Greatest Common Measure of the common measure so found and the third." This Rule may be extended to any number of quantities. Ex. XXXIV. Example worked out. Ex. Find the G. c. M. of x-a1, x3+a3, and x2 — a2. x-a=(x2-a2) (x2+a3), therefore 2-a2 is the G. C. M. of x4 — a1 and x2 — a2. Next to find the G. c. M. of x2-a2 and x3 + a3. x2-a2=(x+a)(x-a), x2+a3=(x+a) (x2 — ax+a2), therefore x+a is the G. c. M. required. Ex. XXXIV. Examples for Practice. Find the G. C. M. of 1. a3+a2b-ab2 — b3, a3— 3ab2+263, and a3-2a2b— ab2+2b3. 2. x2-2α2-ax, x2-4a2, x2-6a2+ax, and x2-8a2+2ax. 3. a3+b3, a1-ba, a3 + b3. 4. 3x-7x2y+5xy-y3, x2y+3xy3-3x3-y3, and 3x+5x3y+xy3-y3. THE LEAST COMMON MULTIPLE. 75. DEF. If any number of algebraical expressions be arranged according to descending powers of some common letter, then the quantity of the lowest dimensions with respect to that letter, which is divisible by each of the expressions, is called the LEAST COMMON MULTIPLE (L.C.M.) of the expressions. 76. To investigate a Rule for finding the Least Common Multiple of two algebraical expressions. Let A and B be the expressions; let D be their Greatest Common Measure, so that A=mD and B=nD; |