III. To Reduce fractions in Algebra to their lowest terms, cancel out any factors common to numerator and denominator as in arithmetic; when the quantities readily split into factors, this can be done at sight; in more difficult cases find the G.C.M. of the numerator and denominator, and cancel this out of each, -2 (7.) x3+10x2+25x-28 and x2+14x+49. Ans. x+7 (8.) x3-9x2+15x-2 and x2-4. Ans.x-2. (9.) a2+4a-5, a2+7a+10 and a2-5. Ans. a +5. (10.) a3+1 and a3 +ba2+ab+1. Ans. a+1. (11.) a2-3x+2, a2 —a— -2 and a2-4a +4. Ans. a— (12.) b+by-y2-y3 and b2+by+b+b3. Ans. y+1 (13.) a2-a-2 and 1+a3. Ans. a+1. (14.) 3a2+4a-4 and 3a2+a-2. Ans. 3a-2. (15.) 8a2+14a-15 and 8a3 +30a2+13a−30a. Ans. 8a2+14a-15. (16.) 6a2+7a-3 and 12a2+16a-3. Ans. 2a+3. (17.) 2a3-10a2+12a aud 3a1—15a3 +24a2—24. Ans. a (18.) a3 +ab+ab2+b3 and a1+a3b+ab3 —b1. -2. Ans, a2+b2. (19.) a1- 2a3b + 2ab3-b1 and a1- 2a3b + 2a2b2— 2ab3+b4. Ans. (a-b) (20.) a3-8a+3 and a6+3a5+a+3. Ans. a +3. (21.) at-qa3-q2a+q3 and ga2 — q3. (22,) 18a3-33a2+44a-35 and 6a3 —19a2+38a-28 Ans. a-q. 23. 12a +29a2+14a and 12a3 -a2-6α. Ans. 6a-7. (26.) 2a5-a4+2a3-10a2-4a-5 and 10a2+5a+5. (27.) 8a4b+2a3b-2a2-3a2b+a and 14a2-7a. Ans. 2a2+a+1. Ans. 2a2-a. (28.) a1-x4 and a3 — a2x-ax2+x3. Ans. a2-x2. LEAST COMMON MULTIPLE.-L.C.M. See Arithmetic. Find the L.C.M. of all the co-efficients. To this affix all the algebraical quantities occurring in all the expressions, with their highest powers. Thus find the L. C. M. of 16x2y2x, 8x3y and 12ya2. Here the L.C.M. of 16, 8, and 12, is 48; and the letters common to all the three expressions are x, y, z, a: so the L.C.M. of the three expressions is 48.xyz u2. Find the L.C.M. of the following expressions: (1.) a2-b2 and as-b3. Ans. a+a3b — ab 3 —b1. (2.) x2-bx-ex+be and x2-ax-ex+ae. Ans. (x-a) (x—b) (x—e.) (3.) 55ay +5xy2-5y3 and 6x3 + 6xy+6xy +by". Ans. 30x4-30y*. (4.) 4-4y2; 12xy2+12ys; 8x3-8x2y. Ans. 24ty2-24x2y1. (5.) 6a2b+6ab2; 9a3-9ab2; 4a3-4ab2. Ans. 36a3b2-36ab1. (6.) a- and a3-a2x-ax2+x3. (7.) x3-a2x-ax2+a3 and a2x2-a. Ans. a5-ax-ax1+x5. x-a1, and ax3 +a3xAns. ax5-a2x4-a5x+a®. Ans. a+ax-ax3-x. (8.) ax3, and a2-x2. (10.) 1-2; 1+2 and 1-x. Ans. x4-5x2+4. To find the L.C.M. of two compound expressions, first find their G.C.M., and divide either of the quantities by it. The product of this quotient and the quantity will be the L.C.M. required. If there are three compound expressions, find the L.C.M. of any two of them, then that of the result, and the third; and so on for any number of expressions. (12.) 3-11x+6, 2x2-7x+3; and 6x2-7x+2. Ans, 6x3-25x2+23x−6. (13.) 6x2-13x+6; 12x2 -5x-2 and 15x2+2x−8. Ans. 120-134x3-129x2+74x +24. (14.) a-b: a2-b2; and a2+2ab+b2. Ans as+a2b — ab2 —b3. › (15.) a+b; a2 —b2 ; and a2 +2ab+b2. Ans. as+a2b-ab2 —b3. (16.) x+2; −4; x2+4x+4. Ans. 3+2x2-4x-8. (17.) x2+4x+4; x−2; x2−4. Ans. x3 +2x2-4x-8. (18.) -9; x2+6x+9; x+3. Ans. x3 +3x2-9x-27. (19.) x-xу; x2 —x; ay—ay2. Ans. ax2y-ax3y-axy+axy3. (20.) x-2x; x2-x; 2a-4a. Ans. 2-5x+6y2. (22.) a-3b; a3-5ab+6b; a-2b; a-b. Ans. a3-6a2b+11ab2+6ba. (23.) 1-y'; 1+y; 1-y. Ans; 1—y2. (24.) x3-y3 and (x+y) (x−y). Ans. x+x3y-xy3—y1. (25.) (y−2) (y+2); y−2; (y−1) (y+1); y-1. Ans. y-5y+4. FRACTIONS. For arithmetical principles involved in dealing with vulgar fractions, see arithmetic. CASE 1.-To express an improper fraction as a mixed quantity; (1.) Express as a mixed quantity. Ans. 2x+ 19. 8 3x 8 as a mixed quantity. За 7 Ans. 2xy +7 (15.) Ans. 2a+6+· a -3 (16.) 5b-y 1062—17by+10y2 Ans. 2b—3y+ 7y 5b-y CASE. To reduce a mixed quantity to an improper fraction proceed as in arithmetic. a -3 3x EXAMPLES.-Reduce 2x+ 8 762 За 3a2+7bs Reduce a+: -to an improper fraction. Ans. За Reduce the following mixed quantities to improper fractions: |