DIVISION.

76. Division, in Algebra, is the process of finding how many times one quantity, called the divisor, is contained in another quantity, called the dividend; the result of division is called the quotient.

It follows, therefore, that the quotient must be a quantity which multiplied by the divisor, will produce the dividend. Thus, reversing the process of multiplication, we have,

abc÷a bc, because bc>a=abc

77. It was shown in the multiplication of monomials, (66), that the coefficient of the product is found by multiplying together the coefficients of the factors; and that the exponent of any letter in the product is found by adding together the exponents of this letter in the factors. Hence, in division,

1.—The coefficient of the quotient must be found by dividing the coefficient of the dividend by that of the divisor; and

2.--The exponent of any letter in the quotient must be found by ⚫ subtracting the exponent of this letter in the divisor from its exponent in the dividend. Thus,

24a°÷6a24a5-3-4a'

It was shown in multiplication, (66), that when two factors have like signs, their product is positive; and that when two factors have unlike signs, their product is negative. In division, therefore, when the dividend is positive, the quotient must have the same sign as the divisor; and when the dividend is negative, the quotient must have the sign unlike that of the divisor. And there will be four cases,

3.-If the dividend and divisor have like signs, the quotient will be positive; but if the dividend and divisor have unlike signs, the quotient will be negative.

CASE I.

78. When the divisor is a monomial.

From the principles already given we have the following
RULE. To divide one monomial by another;-

I. Divide the coefficient of the dividend by the coefficient of the divisor, for a new coefficient.

II. To this result annex the letters of the dividend, with the exponent of each diminished by the exponent of the same letter in the divisor, suppressing all letters whose exponents become zero.

III. If the signs of terms are alike, prefix the plus sign to the quotient; if they are unlike, prefix the minus sign.

To divide a polynomial by a monomial ;—

Divide each term of the dividend separately, and connect the quotients by their proper signs.

NOTE.-It may happen that the dividend will not exactly contain the divisor; in this case the division may be indicated, by writing the dividend above a horizontal line, and the divisor below, in the form of a fraction. The result thus obtained may be simplified, by suppressing all the factors common to the two terms; thus,

4x2yz2+6x2 y2 z=

4x2yz2 2z
6x2y22 3y

But as this process is essentially a case of reduction of fractions, we shall omit such examples till the subject of fractions is reached.

11. Divide 34ay" by -17xy. 14. Divide (a-c)' by (a—c)".

13. Divide 35(x+y)3 by 5(x+y).

Ans. -2x-1y-1.
Ans (a—c).

Ans. 7(x+y).

Ans. d+4x-36.

14. Divide 12m3d(c-x)" by 3md (c-x3)". 15. Divide 3bcd+12bcx-96'c by 3bc. 16. Divide 15a2bc-15acx2+5ad'c by -5ac. 17. Divide 10x3-15x2-25x by 5x. 18. Divide 15-45x+10x3-105x by 5x2. 19. Divide a c-am-1c2+am-2c3-a-cam-c by ac. Ans. a--a2c+am-3c3—am-c3 +am~*c*.

Ans. 2x2-3x-5.

20. Divide 3m2(a—b)2—3m(a—b) by 3(a—b).

Ans, am-bm2-m.

21. Divide 7a(3m—2a)—(3m—2a)1 by (3m—2a).

Ans. 9a-3m.

CASE II.

79. When the divisor is a polynomial.

Suppose both dividend and divisor to be arranged according to the descending powers of some letter. Then it follows, from (75, 1), that the first term of the dividend must be the product of the first term of the divisor by the first term of the quotient similarly arranged. We can therefore obtain this term of the quotient, by simply dividing the first term of the dividend by the first term of the divisor, thus arranged. The operation may then be continued in the manner of long division in Arithmetic; each remainder being treated as a new dividend, and arranged as the first. 1. Divide 6a+ab-20a b'+17ab'-46' by 2a-3ab+b2.

OPERATION.

6aab-20a'b'+17ab'-4b

6a-9ab+3a2b2

10a b-23a2b+17ab

10a b-15a2b2+ 5ab

-8a'b'+12ab-4b

-8a'b'+12ab' 46'

12a-3ab+b, Divisor.

3a'+5ab-4b', Quotient

Hence we have the following

RULE. I. Arrange both dividend and divisor according to the descending powers of one of the letters.

II. Divide the first term of the dividend by the first term of the divisor, and write the result in the quotient.

III. Multiply the whole divisor by the quotient thus found, and subtract the product from the dividend.

IV. Arrange the remainder for a new dividend, with which proceed as before, till the first term of the divisor is no longer contained in the first term of the remainder.

V. Write the final remainder, if there be any, over the divisor in the form of a fraction, and the entire result will be the quotient sought.

EXAMPLES FOR PRACTICE.

Ans. a'+2ax+x".
Ans. a3-3ac+c.

1. Divide a'+3a2x+3ax2+x' by a+x.
2. Divide a3-4a2c+4ac-c3 by a-c.
3. Divide a-6a2+12a-8 by a2-4a+4.
4. Divide 3x-2x*x*—x3-2x-15 by x3-5-4x.

5. Divide 25-x-2x-8x' by 5x-4x'.

6. Divide 6a+9a-15a by 3a2-3a.

Ans. a-2.

Ans. x-2x+3.

Ans. 5x+4x+3x+2.
Ans. 2a+2a+5.

7. Divide xy" by x+2xy+2xy+y3.

Ans. x-2xy+2xy3—y'

3. Divide ax-(a+b).x2+b2 by ax-b. Ans. x2-ax-b

9. Divide a⭑+46' by a2-2ab+263.

Ans. a'+2ab+2*

10. Divide x-x1+x3—x2+2x-1 by x2+x-1.

11. Divide 1+3x by 1-5x. Ans. 1+8x+40x2+200x3+etc.

12. Divide 1-x-x2 by 1+x+x3.

Ans. 1—2x+2x3—2x*+2x*—2x2+2xo—2x1o+etc.

13. Divide x-2x+1 by x2-2x+1.

Ans. x+2x+3x2+2x+1.

14. Divide a+b+c-3abc by a+b+c.

Ans. a+b+c-bc-ac-ab.

15. Divide 2x'y-5x'y'-11x'y'+5x*y*-26x'y' +7x'y°-12xy' by x-4x3y+x'y'—3xy". Ans. 2xy+3x'y'—xy3+4y'.

16. Divide a+c+a+c*—a*c-ac-2a'c' by a'+c-a'c-ac2.

Ans. a2+c2+a+c.

17. Divide 4x-5x+8x-10x-8x2-5x-4 by 4x+3x+2x

+1.

Ans. x3-2x+3x-4.

Ans. x+x3a+x3a2+xa3+a*.

Ans. a2+ax+x2+.

2x3

18. Divide x-ao by x—a. 19. Divide a +x3 by a—x. 20. Divide x-xym1_xm—1y+y” by x—y. "1y+yTM by x-y. Ans. x-ym-1. 21. Divide ao+m_a©b”—amba+b”+d by am—b”. Ans. a-ba. 22. Divide x2-2x3y"—2x”y3+y3 by x2+y”.

-1

Ans. x-3xy”+y”.

EXACT DIVISION.

80. Division is said to be exact when the quotient contains no fractional part; the quotient in this case is said to be entire.

81. It follows from the rule of division, (78), that the exact division of one monomial by another will be impossible under the following conditions:

1. When the coefficient of the divisor is not exactly contained in the coefficient of the dividend.

2.-When a literal factor has a greater exponent in the divisor than in the dividend.

3. When a literal factor of the divisor is not found in the dividend.

82. It is also evident, from (79), that the exact division of one polynomial by another will be impossible,

1. When the first term of the divisor arranged with reference to any one of its letters, is not exactly contained in the first term of the dividend arranged with reference to the same letter.

2.—When a remainder occurs, having no term which will exactly contain the first term of the divisor.