39. If two terms of a trinomial are squares, and the third term is equal to twice the product of their square roots, the trinomial may be factored by means of principle 1°, Art. 35.

8. Factor a2 + 2ab + b2.
9. Factor 4x2 + 12xy + 9y2.

10. Factor x2 + 12x + 36. 11. Factor 4x4 + 4x2y + y2.

Ans. (a + b) (a+b).

Ans. (2x + 3y) (2x + 3y).
Ans. (x+6) (x + 6).

Ans. (2x2 + y) (2x2 + y).

12. Factor 4a2b2 + 12abc + 9c2.

Ans. (2ab+3c) (2ab+3c).

13. Factor 16a1y1 + 8a2y1z2 + y1z1.

Ans. (4a2y2+ y2x2) (4a2y2 + y2x2).

40. If two terms of a trinomial are squares, and the third term is equal to minus twice the product of their square roots, the trinomial may be factored by means of principle 2°, Art, 35.

18. Factor 4x2-4x2y + y2. Ans. (2x2-y) (2x2—y).

19. Factor 36x2-24xy+4y2.

Ans. (6x 2y) (6x — 2y).

20. Factor 4x2y2 — 4xyz + z2.

Ans. (2xyz) (2xy — z).

41. If the two terms of a binomial are squares, and have contrary signs, the binomial may be factored by means of principle 3°, Art. 35.

The following examples may be factored by means of formula 4°, Art. 35:

27. x2 + 13x + 42 = x2 + (6 + 7)x + 6 × 7

= (x + 6) (x + '7).

x

28. x2+ 2x-15= x2 + (5-3)x - 3 × 5

29. x2-15x+56= x2 - (7 + 8)x−7 × -8

—

The following examples may be factored by means of formulas 5o, 6o, 7°, and 8°, Art. 35:

31. 8a3 b3

=

(2a — b) (4a2 + Qab + b).

32. a3 + 64m3 = (a + 4m) (a2 - 4am + 16m2). 33. 16a36a2b2 + 81b4

=

(4a2 + 6ab +962) (4a2 — 6ab + 963).

34. a4b481c4 = (a2b2 + 9c2) (a2b2—9c2)

Ans. a2 (bc)2 = (a + b − c) (a − b + c).

Ans. a4-(3ab+c2)2 = (a2+3ab+c2) (a2—3ab—c2).

CHAPTER III.

GREATEST COMMON DIVISOR, AND LEAST COMMON MULTIPLE.

I. GREATEST COMMON DIVISOR.

Definitions.

42. A common divisor of two quantities, is a quantity that will divide both without a remainder. Thus, 3ab, is a common divisor of 9a2b2c and 3a2b2—6a3b3.

43. A simple or prime factor is one that cannot be resolved into any other factors.

Every prime factor, common to two quantities, is a common divisor of those quantities. The continued product of any number of prime factors, common to two quantities, is also a common divisor of those quantities.

44. The greatest common divisor of two quantities, is the continued product of all the prime factors that are common to both.

It is called the greatest common divisor, because it is greater with respect to its coefficients and exponents than any other common divisor.

There are two methods of finding the greatest common divisor: by factoring, and by continued division.

Method by Factoring.

45. If both quantities can be resolved into prime factors by the methods already given, the greatest common divisor may be found by the following

RULE.

I. Resolve both quantities into prime factors.

II. Find the continued product of all the prime factors common to both; it will be the greatest common divisor required.

EXAMPLES.

1. Let it be required to find the greatest common divisor of 42abx and 70αcx:

Factoring, we have,

42abx 7a × 2x × 3b,
70acx=7a × 2x × 5c.

The factors 7a and 2x are common; hence, the greatest common divisor is 7a × 2x, or 14ax.

2. Find the greatest common divisor of 3ax2 + 3x3 and 2ay + 2xy:

hence, the greatest common divisor is a + x.

3. Find the greatest common divisor of 2a3 — 4a2b +2ab2 and 2a3 2ab2: