EXAMPLES X. b.
Resolve into factors :
1. a^ + ab + ac+ bc. 12. 3 ax — bx — 3 ay + by. 2. a^2 - ac + ab - bc. 13. 3x^2 + 3xy — 2ax- ay. 3. a^c^ + acd + abc + bd. 14. mx — 2 my — 7nx+ 2 ny. 4. a^ +3a + ac + 3c. 15. ax^2 -3bxy - axy + 3 by^2. 5. 2 x+ elxx + 2 c + c^2. 16. x^ + mxy — 4xy — 4 my^. 6. x^ - ax + 5 x - 5 a. 17. 2 x^ - x^ + 4 x - 2. 7. 5a + ab + 5b+ b^. 18. 3x^3 + 5x^2 + 3x + 5. 8. ab -by-ay + by. 19. x^ + x^ + 2 x + 2. 9. mx — my — nx + ny. 20. y^ — y^ + y — 1. 10. mx — ma + nx — na. 21. axy + bcxy — az — bcz. 11. 2 ax + ay + 2 bx + by. 22. f^x^ + g^x^ - ag^ - af^.
TRINOMIAL EXPRESSIONS.
92. When the Coefficient of the Highest Power is Unity. Before proceeding to the next case of resolution into factors the student is advised to refer to Chap. iv. Art. 51. Attention has there been drawn to the way in which, in forming the product of two binomials, the coefficients of the different terms combine so as to give a trinomial result. Thus, by Art. 51,
(x + 5)(x + 3)=x^+ 8x+15 .... (1), (x-5)(x-3)=x^-8x+15 .... (2), (x+5)(x-3)=x^+2x-15 .... (3), (x - 5)(x + 3)= x^2- 2x-15 .... (4).
We now propose to consider the converse problem : namely, the resolution of a trinomial expression, similar to those which occur on the right-hand side of the above identities, into its component binomial factors. By examining the above results, we notice that :
1. The first term of both the factors is x. 2. The product of the second terms of the two factors is equal to the third term of the trinomial ; thus in (2) above we see that 15 is the product of — 5 and — 3 ; while in (3) — 15 is the product of + 5 and — 3.
