EXAMPLES X. a.
Resolve into factors :
1. a^ -ax. 5. 8 x- 2 x 9. 15 a ^- 225 a ^ 2. x^ - x^ 6. 5 cx - 5 a^x^ 10. 54 - 81 x. 3. 2 a - 2 a^2. 7. 15 + 25 x^. 11. 10 x^ - 25 x^y. 4. 7p^2 + p. 8. 16 x + 64 x^j. 12. 3 x^ - x^2 + x. 13. 3a^-3a^b-6a^b^. 16. 5 x^ - 10 a^x^ - 15 a^x^ 14. 2 x^2y^3 - 6 x^2y^2 + 2 xy^3. 17. 7 a - 7 a^ + 14 a^4, 15. 6 x^3 - 9 x^2y + 12 xy^2. 18. 38 a^x^ + 57 a^x^2.
WHEN THE TERMS CAN BE GROUPED SO AS TO CONTAIN A COMMON FACTOR.
91. Ex. 1. Resolve into factors x^2 — ax + bx — ab.
Noticing that the first two terms contain a factor x, and the last two terms a factor b, we enclose the first two terms in one bracket, and the last two in another. Thus,
x^2 — ax + bx — ab = (x^2 — ax) + (bx — ab) = x(x — a) + b(x— a) . . . (1) = (x- a)(x + b).
since each bracket of (1) contains the same factor x — a.
Ex. 2. Resolve into factors 6 x^2 — 9 ax + 4 bx — 6 ab.
6 x^2 - 9 ax + 4 bx - 6 ab = (6 x^2 - 9 ax) + (4 bx - 6 ab) = 3 x(2 x - 3 a) + 2 b(2 x - 3 a) = (2x-3a)(3x + 2b).
Ex. 3. Resolve into factors 12 a^2 — 4ab — 3 ax^2- + bx^2.
12 a^2-4ab-3 ax^2 + bx^2 = (12 a^2 - 4 ab) - (3 ax^2 - bx^2) = 4a(3a -b)-x^2(3a- b) = (3a-b)(4a-x^2).
Note. In the first line of work it is usually sufficient to see that each pair contains some common factor. Any suitably chosen pairs will bring out the same result. Thus, in the last example, by a different arrangement, we have
12 a^2 - 4 ab - 3 ax^2 + bx^2 =(12 x^2 - 3 ax^2)- (4 ab - bx^2) = 3a(4a - x^2)-b(4a-x^2) = (4a-x^2)(3a-b).
The same result as before, for the order in which the factors of a product are written is of course immaterial.
