CHAPTER III.

FACTORING, GREATEST COMMON DIVISOR, AND

LEAST COMMON MULTIPLE.

USEFUL FORMULAS.

53. A formula is an algebraic expression of a general rule or principle.

Formulas serve to shorten algebraic operations, and are also of much use in the operation of factoring. When translated into common language, they give rise to practical rules.

The verification of the following formulas affords additional exercises in multiplication and division.

54. Formula 1.-To form the square of a + b, we have (a+b)2 = (a+b)(a+b) = a2+2ab+b2 = a2 + b2 + 2 ab; that is,

The square of the sum of any two quantities is equal to the sum of their squares, plus twice their product.

(1) Find the square of 2a+3b.

We have from the rule,

(2a+3b)2 = 4a2 + 9b2 + 12 ab. (2) Find the square of 5 ab+3ac.

NOTE.. If the expression to be squared consists of more than two terms, it may be written in the form of a binomial, and squared. Thus,

(a+b+c)2 = [a + (b + c)]2 = a2 + (b + c)2 + 2 a (b + c)

= a2 + b2 + c2 + 2 ab + 2ac + 2bc;

(a + b + c + d)2 = [(a + b) + (c + d)]2

=

· a2 + b2 + c2 + d2 + 2 ab + 2 ac + 2 ad + 2 bc + 2bd + 2 cd.

Find the square of the following:

(5) 2a+3b+c.

-

55. Formula 2. To form the square of a difference, ab, we have

(a - b)2 = (a - b)(a - b) = a2 — 2ab+b2 = a2 + b2 — 2ab; that is,

The square of the difference of any two quantities is equal to the sum of their squares, minus twice their product.

(1) Find the square of 2a - b.

and

Ans. 49ab144 a2b-168 a3b5.

(ab+c) [a− (b − e)]2 = a2 + (b −c)2 - 2a (b −c)

=

· a2 + b2 + c2 −2ab+2ac2bc;

(a-b-c-d)2= [(a - b) - (e + d)]2

=

(ab)2+(cd)2 - 2 (a - b)(c + d)

· a2 + b2 + c2 + d2 - 2 ab - 2 ac - 2 ad + 2bc+2bd + 2 cd.

56. Formula 3. To multiply a + b by ab, we have

that is,

(a + b) x (a - b) = a2 — b2 ;

The sum of two quantities, multiplied by their difference, is equal to the difference of their squares.

(1) Multiply 2c+b by 2c-b.

(2) Multiply 9ac3 bc by 9ac-3 bc.

(3) Multiply 8a3+7ab2 by 8a3 — 7 ab2.

Ans. 4cb2.

Ans. 81 a'c' 9 b2c3.

Ans. 64 a- 49 a2b1.

NOTE. To multiply a + b c by a − b + c, we have

(a + b − c) × (a -- b + c) = [a + (b − c)] × [a − (b − c)] = a2 − (b − c)3.

Multiply together the following:

(4) x+y-4 and x-y+4.

(5) 2a-b-d and 2a+b+d.

57. Formula 4. — To multiply a2+ab+b2 by a-b, we have (a2 + ab + b2)(a — b) = a3 — b3.

58. Formula 5. To multiply a2 - ab+b2 by a+b, we have (a2 — ab + b2)(a + b) = a3 + b3.

59. Formula 6.To multiply together a+b, a-b, and a2+b2, we have

(a + b)(a − b)(a2 + b2) = a* — b1.

60. Since every product is divisible by any of its factors, each formula establishes the principle set opposite its number.

(1) The sum of the squares of any two quantities, plus twice their product, is divisible by their sum.

(2) The sum of the squares of any two quantities, minus twice their product, is divisible by the difference of the quantities.

(3) The difference of the squares of any two quantities is divisible by the sum of the quantities, and also by their difference.

(4) The difference of the cubes of any two quantities is divisible by the difference of the quantities; also by the sum of their squares, plus their product.

(5) The sum of the cubes of any two quantities is divisible by the sum of the quantities; also by the sum of their squares minus their product.

(6) The difference between the fourth powers of any two quantities is divisible by the sum of the quantities, by their difference, by the sum of their squares, and by the difference of their squares.

FACTORING.

61. Factoring is the operation of resolving a quantity into factors. The principles employed are the converse of those of multiplication. The operations of factoring are performed by inspection. Thus,

(1) What are the factors of the polynomial ac+ab+ ad?

We see by inspection that a is a common factor of all the terms: hence it may be placed without a parenthesis, and the other parts within. Thus, ac + ab + ad = a (c + b + d).

(2) Find the factors of the polynomial a2b2 + a2d — a2f.

Ans. a3 (b2+d-ƒ).

(3) Find the factors of the polynomial 3a2b-6a2b2 + b2d. Ans. b (3a2 — 6a2b+bd).

(4) Find the factors of 3a2b-9a2c - 18 a2xy.

Ans. 3a (b-3c-6xy).

(5) Find the factors of 8a cx-18 acx2+2ac3y — 30 aoc3. Ans. 2ac (4ax-9x2+c1y-15a3c3).

(6) Factor 30 a1b3c — 6 a3b3d3 +18 a3b2c2.

62. When two terms of a trinomial are squares, and positive, and the third term is equal to twice the product of their square roots, the trinomial may be resolved into factors by Formula 1. Thus,

Factor the following:

(1) a2+2ab+b2.

(2) 4a2+12ab+962.

(3) 9a2+12ab+4b2.

(4) 4x2+8x+ 4.

(5) 9a2b2+12 abc +4 c2.

(6) 16x'y2+16xy3+4y1.

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

Ans. (2a+3b)(2a+3b).

Ans. (3a+2b)(3a+2b).

Ans. (2x+2)(2x+2).

Ans. (3ab2c)(3 ab+2c). Ans. (4xy+2y2)(4xy+2y2).

NOTE. Factor the following by note to Formula 1.

(7) a2+2ab+2ac + b2 + c2 + 2bc = a2 + b2 + c2+2ab Ans. (a+b+c)(a+b+c).

+2ac+2bc.

(8) x2+4a2+9b2+4 ax+6 bx+12 ab.

Ans. (x+2a+3b)(x+2a+3b).

63. When two terms of a trinomial are squares, and positive, and the third term is equal to minus twice their square roots, the trinomial may be factored by Formula 2. Thus,