product of the two parts. N. B. The product of the two parts will be plus or minus, according to the sign between the binomial.

Let us now examine the product of a+b into a-b.

Thus, by inspection we find the product of the sum and difference of two quantities is equal to the difference of their squares.

The propositions included in this article are proved also in geometry.

(Art. 14) We can sometimes make use of binomial quantities greatly to our advantage, as a few of the following examples will show:

1. Multiply a+b+c or square the trinomial quantity a+b+c. Suppose a+b be represented by s, then it will be sc. The square of this is s2-2sc+c2, restoring the value of s, and we have (a+b)2+2(a+b)c+c2

2. Square x+y-z. Let x+y=s.

Then (s-2)2=s2-2sz+z2=(x+y)2-2(x+y)x+23. 3. Multiply x+y+z by x+y-z. Ans. (x+y)2-22. 4. Multiply 2x2 - 3x+2 by x-8.

Ans. 2x3-19x2+26x-16.

5. Multiply ax+by by ax+cy.

"Ans. a2x2+(ab+ac)xy+cby.

6. Multiply x+y by x-y.

Ans. Ax2-y2.

DIVISION.

(Art. 15.) Division is the converse of multiplication, the product being called a dividend, and one of the factors a divisor. If a multiplied by b give the product ab, then ab divided by a must give b for a quotient, and if divided by b, give a. In short, if one simple quantity is to be divided by another simple quantity, the quotient must be found by inspection as in division of numbers.

In this last example, and in many others, the absolute division cannot be effected. In some cases it can be partially effected, and the quotients must be fractional.

5. Divide 3acx2 by acy.

6. Divide 7262x by Sabx.

7. Divide 27aby by 11abx.

3x2

Ans.

(Art. 16.) It will be observed that the product of the divisor and quotient must make the dividend, and the signs must conform to the principles laid down in multiplication.

The following examples will illustrate :

8. Divide -9y by 3y.

9. Divide -9y by -3y.

10. Divide +9y by -3y.

Ans. -3.

Ans. +3.

Ans. -3.

(Art. 17.) The product of a3 into a2 is a5, (Art. 10,)

that is, in multiplication we add the exponents, and as division is the converse of multiplication, to divide powers of the same letter, we must subtract the exponent of the divisor

(Art. 18.) The object of this article is to explain the

nature of negative exponents.

Divide a successively by a, and we shall have the fol

Divide a again, rigidly adhering to the principle that to

divide any power of a by a, the exponent becomes one less, and we have

a3, a2, a, a, a, a-2, a-3, &c.

Now these quotients must be equal, that is, a3 in one

series equals a3 in the other, and

1

1

a

a2= a2, a=a', 1=a°, ==a-a-2

1

Another illustration. We divide exponential quantities by subtracting the exponent of the divisor from the exponent of the dividend. Thus as divided by a2 gives a quotient of a 5-2=a3. a5 divided by a7 =a5-7=a-2. We can also divide by taking the dividend for a numerator and

From this we learn, that exponential terms may be changed from a numerator to a denominator, and the re

verse by changing the signs of the exponents.

Observe, that to divide is to subtract the exponents.

7bd

7

Divide 14ab2cd by 6a2bc2. Ans. 3ac3bda-c.

(Art. 19.) A compound term divided by a simple term, is effected by dividing each term of the compound quantity by the simple divisor.

EXAMPLES.

1. Divide 3ax-15x by 3x.

2. Divide 8x3+12x2 by 4x2.

Ans. a-5.

Ans. 2x+3.

3. Divide 3bcd+12bcx-9b2c by 3bc. Ans. d+4x-36.

5. Divide 15a2be-15acx2+5ad2 by -5ac.

d

6. Divide 10x3-15x3-25x by 5x.

Ans. -3ab3x2이

Ans. 2a -3x-5.

7. Divide -10ab+60ab3 by-bab.

5

Ans. -1062.

3

8. Divide 36a2b2+60a2b-6ab by -12ab.

Ans. -3ab-5a+

9. Divide 10rx-cry+2crx by cr.

10. Divide 10vy+16d by 2d.

11. Divide 6ay-18acd+24a by 6a.

12. Divide mx-amx+m by m.

(Art. 20.) We now come to the last and most important operation in division, the division of one compound quantity by another compound quantity.

The dividend may be considered a product of the divisor into the yet unknown factor, the quotient; and the highest power of any letter in the product, or the now called dividend, must be conceived to have been formed by the highest power of the same letter in the divisor into the highest power of that letter in the quotient. Therefore, both the divisor and the dividend must be arranged according to the regular powers of some letter.

After this, the truth of the following rule will become obvious by its great similarity to division in numbers. RULE. Divide the first term of the dividend by the first term of the divisor, and set the result in the quotient.* Multiply the whole divisor by the quotient thus found, and subtract the product from the dividend.

The remainder will form 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.

The divisor and remainder, if there be a remainder, are then to be written in the form of a fraction, as in division of numbers.

EXAMPLES.

Divide a2+2ab+b2 by a+b.

Here, a is the leading letter, standing first in both dividend and divisor, hence no change of place is necessary.

*Divide the first term of the dividend and of the remainders by the first term of the divisor; be not troubled about other terms.