3. The algebraic sum of the second terms of the two factors is equal to the coefficient of x in the trinomial ; thus in (4) the sum of — 5 and + 3 gives — 2, the coefficient of x in the trinomial.
In showing the application of these laws we will first consider a case where the third term of the trinomial is positive.
Ex. 1, Resolve into factors x^2 + 11 x+ 24.
The second terms of the factors must be such that their product is + 24, and their sum + 11. It is clear that they must be + 8 and + 3.
.-. x^2 + x + 24=(x + 8)(x + 3).
Ex. 2. Resolve into factors x^ —10x+ 24.
The second terms of the factors must be such that their product is + 24, and their sum — 10. Hence they must both be negative, and it is easy to see that they must be — 6 and — 4.
x^2- 10x + 24=(x-6)(x-4).
Ex. 3. x^2- x + =(x-9)(x-9) = (x-9)^2.
Ex. 4. x^ + 10 x^ + 25 = (x^2 + 5) (x^ + 5) = (x^ + 5)^2.
Ex. 5. Resolve into factors x^ — 11 ax + 10 a^2.
The second terms of the factors must be such that their product is + 10 a^2, and their sum — 11 a. Hence they must be — 10 a and — a.
x^2 — 11 ax + 10 a^2 =(x — 10 a) (x — a).
Note. In examples of this kind the student should always verify his results, by forming the product (mentally, as explained in Chapter IV.) of the factors he has chosen.
EXAMPLES X. c.
Resolve into factors :
1. a^2 + 3 a + 2. 7. x^2 -21x + 108. 13. x^2 + 20x + 96. 2. a^2 + 2a + 1. 8. x^2 - 21x + 80. 14. x^2-26x+165. 3. a^2 + 7 a + 12. 9. x^2 + 21x + 90. 15. x^2-21x + 104. 4. a^2 - 7 a + 12. 10. x^2 -19x + 84. 16. a^2 + 30a + 225. 5. x^2-11x + 30. 11. x^2 -19x + 78. 17. a^2 + 54a + 729. 6. x^2- 15 x + 56. 12. x+2 -18x + 45. 18. a^2- 38a + 361. 19. a^2 - 14 ab + 49b^2 23. x^2 -23xy + 132y^2. 20. a^ + 5ab- 6b^2. 24. x^2 - 26xy+ 169y^2. 21. m^2 - 13mn + 40 n^2. 25. x^4 + 8 x^2 + 7. 22. m^2 - 22mn + 105 n^2. 26. x^4 + 9 x^2y^2 + 14 y^2.
