Page:Elementary algebra (1896).djvu/93

75 Rh 13. x^ + x- 110. 18. x^y^ - 5xy - 24. 23. a^ - a - 26. 14. x^-9x- 90. 19. x^2 + ax - 42 a^. 24. a^y^ + ay-240. 15. x^-x- 240. 20. x^2-32 xy-105 y^2 25. a^b^ - a^b^ - 56b^4. 16. a^2 - 12 a - 85. 21. x^2 + 18 x - 115. 26. x^ - 14 x^2 - 51. 17. a^ - 11a - 152. 22. x^2 + 16x - 260. 27. y^+6 x^y^-27 x^. 28. a^ + 12 abx - 28 b^x^. 31. x^ - a^x^ - 132 a^4. 29. a^2 - 18 axy - 243 x^y^ 32. x^ - a^x^ - 462 a^. 30. x^ + 13 a^x^ - 300 a^w 33. x^ + x^ - 870. 34. 2 + x - x^ 36. 110 - x - x^. 38. 120 - 7 ax - a^x^. 35. 6 + x - x^2. 37. 380 - x - x^.

94. When the Coefficient of the Highest Power is not Unity. Again, referring to Chap. iv. Art. 51, we may write the following results:

(3x + 2)(x+4)=3x^ + 14x - 8. . . (1), (3x-2)(x-4)=3x^- x + 8. . . (2), (3x+2)(x-4)=3x^ - 10x - 8. . . (3), (3x-2)(x + 4)=3x^ +10x - 8. . . (4).

The converse problem presents more difficulty than the cases we have yet considered.

Consider the result 3 x-14x + 8 = (3x- 2) (x - 4).

The first term 3 x^ is the product of 3 x and x.

The third term + 8 is the product of — 2 and — 4.

The middle term — 14 x is the result of adding together the two products 3x — 4 and x  —2.

Again, consider the result 3 x^— 10 x—8 = (3 x + 2)(x— 4).

The first term 3 x^ is the product of 3 x and x.

The third term — 8 is the product of + 2 and — 4.

The middle term — 10 x is the result of adding together the two products 3 x — 4 and x 2; and its sign is negative because the greater of these two products is negative.

Considering in a similar manner results (1) and (4), we see that :

1. If the third term of the trinomial is positive, then the second terms of its factors have both the same sign, and this sign is the same as that of the middle term of the trinomial.