Page:Elementary algebra (1896).djvu/122

104 Rh 20. Sx^ + Ux + e, 3 :c2 + 8 X + 4, x- + 5 .r + 6. 21. 5 x2 + 11 ic + 2, 5 X- + 10 X + 3, :*;- + b x + 6. 22. 1 + x^ (1 + a;)--2, 1 + x^. 23. 2 a:2 + 3 X - 2, 2 x- + 15 x - 8, x^ + 10 x + 16. 24. 3 x2 - X - 14, 3 x2 - 13 X + 14, x"^ - 4. 25. 12 x2 + 3 X - 42, 12 x^ + 30 x^ + 12 x, 32 x^ - 40 x - 28. 26. 3 X* 4- 26 x3 + 35 x^, 6 .x^ + 38 x - 28, 27 x^ + 27 x^ - 30 x. 27. 60 X* + 5 x3 - 5 x2, 60 x'^ij + 32 xy + 4 ?/, 40 x^^ - 2 x^y - 2 x?/. 28. 8 X? - 38 x?/ + 35 ?/"', 4 x^ - x?/ - 5 ?/2, 2 x^ - 5 x?/ - 7 /-, 29. 12 x2 - 23 xy + 10 i/^, 4 x2 - 9 x?/ + 5 y^, 3 x2 - 5 x?/ + 2 ?/2. 30. 6 ax3+7 «2j;2_3 ^^3^;, 3 rt-2j;2+ 14 o^x - 5 «^ 6 x2+39 «x+45 «2, 31. 4 ax'V" + 11 axy- — 3 ay-, 3 x^^=^ + 7 x-^=^ — 6 x^^, 24 ax- — 22 ax + 4 a.

129. L. C. M. of Compound Expressions which cannot be factored by Inspection. When the given expressions are such that their factors cannot be determined by inspection, they must be resolved by finding the highest common factor.

Ex. Find the lowest common multiple of 2 x4 + x3 - 20 x2 - 7 x + 24 and 2 x^4 + 3 x^3 - 13 x2 - 7 x + 15.

The highest common factor is x2 + 2 x — 3.

By division, we obtain 2 x4 + x3 - 20 x2 - 7 x + 24 =(x2 + 2 x - 3)( x2 - 3 x - 8). 2 x4 + 3 x3 - 13 x2 - 7 x + 15 = (x^2 + 2 x - 3) (2 x2 - x - 5).

Therefore the L. C. M. is (x2 + 2 x-3)(2 x2 - 3 x-8)(2 x2-x-5).

130. We may now give the proof of the rule for finding the lowest common multiple of two compound algebraic expressions.

Let A and B be the two expressions, and F their highest common factor. Also suppose that a and b are the respective quotients when A and B are divided by F then A = aF, B = bF. Therefore, since a and b have no common factor, the lowest common multiple of A and B is abF, by inspection.