Page:Elementary algebra (1896).djvu/92

74 Rh 27. x^+49xy + 600y^. 33. 20 + 9x + x^2. 28. x^2y^2 + 34 xy + 289. 34. 132 - 23 x + x^2. 29. a^b^ + 37 a^b^ + 300. 35. 88 + 19 x + x^. 30. a^2 - 29 ab + 54 b^2. 36. 130 + 31 x^ + x^y^. 31. x^ + 162 x^2 + 6561. 37. 204 - 29 x^2 + x^. 32. 12 - 7 x + x^. 38. 216 + 35 x + x^2.

93. Next consider a case where the third term of the trinomial is negative.

Ex. 1. Resolve into factors x^2 + 2 x — 35.

The second terms of the factors must be such that their product is — 35, and their algebraic sum + 2. Hence they must have opposite signs, and the greater of them must be positive in order to give its sign to their sum.

The required terms are therefore + 7 and — 5.

x^2 + 2 x - 35 = (x + 7) (x - 5).

Ex. 2. Resolve into factors x^2 — 3 x — 54.

The second terms of the factors must be such that their product is — 54, and their algebraic sum — 3. Hence they must have opposite signs, and the greater of them must be negative in order to give its sign to their sum.

The required terms are therefore — 9 and + 6.

x^2 - 3x- 54= (x-9)(x + 6).

Remembering that in these cases the numerical quantities must have opposite signs, if preferred, the following method may be adopted.

Ex. 3. Resolve into factors x^2y^2 + 23 xy - 420.

Find two numbers whose product is 420, and whose difference is 23. These are 35 and 12 ; hence inserting the signs so that the positive may predominate, we have

x^2y^2 + 23 xy - 420 = (xy + 35) (xy - 12).

EXAMPLES X. d.

Resolve into factors :

1. x^2 - x - 2. 5. x^2 + 2x - 3. 9. a^2 + a - 20. 2. x^2 + x - 2. 6. x^2 + x - 56. 10. a^2 - 4 a - 117. 3. x^2 - x - 6. 7. x^2 - 4x - 12. 11. x^2 + 9 x - 36. 4. x^2 - 2x - 3. 8. a^2 - a - 20. 12. x^2 + x - 156.