Note. In the multiplication of compound quantities, it is the best way to set them down in order, according to the powers and the letters of the alphabet. And in multiplying them, begin at the left-hand side, and multiply from the left hand towards the right, in the manner that we write, which is contrary to the way of multiplying numbers. But in setting down the several products, as they arise, in the second and following lines, range them under the like terms in the lines above, when there are such like quantities; which is the easiest way for adding them up together.

In many cases, the multiplication of compound quantities is only to be performed by setting them down one after another, each within or under a vinculum, with a sign of multiplication between them. As (a + b) x (a−b) × 3ab, or a + b. a - b. 3ab.

EXAMPLES FOR PRACTICE.

1. Multiply 10ac by 2a.

2. Multiply 3a2 - 2b by 3b.

3. Multiply 3a+2b-by 3a-2b.
4. Multiply x2- xy + y2 by x + y.

5. Multiply a3 + ab + ab2 + b3 by a-b.
6. Multiply a + ab + b2 by a2-ab + b2.

Ans. 20a2c.

Ans. 9a2b-6b2.

Ans. 9a-462.
Ans. 3+ y3.

Ans. a4-b+.

7. Multiply 3.2 −2xy + 5 by x2 + 2xy−6. 8. Multiply 3a - 2ax + 5x2 by 3a2-4ax-7x2. 9. Multiply 3x3 + 2x2y2+3y3 by 2x3-3x2y2 + 3y3. 10. Multiply a2 + ab + b2 by a-2b.

DIVISION.

DIVISION in Algebra, like that in numbers, is the converse of multiplication; and it is performed like that of numbers also, by beginning at the left-hand side, and dividing all the parts of the dividend by the divisor, when they can be so divided; or else by setting them down like a fraction, the dividend over the divisor, and then abbreviating the fraction as much as can be done. This will naturally divide into the following particular cases.

CASE

CASE I.

When the Divisor and Dividend are both Simple Quantities,

SET the terms both down as in division of numbers, either the divisor before the dividend, or below it, like the denominator of a fraction. Then abbreviate these terms as much as can be done, by cancelling or striking out all the letters that are common to them both, and also dividing the one co-efficient by the other, or abbreviating them after the manner of a fraction, by dividing them by their common

measure. \

Note. Like signs in the two factors make in the quotient; and unlike signs make ; the same as in multiplication *.

2. Also c÷c === 1; and abx ÷ bxy =

C

3. Divide 16x2 by 8x.

bxy

y

* Because the divisor multiplied by the quotient, must produce the dividend. Therefore,

1. When both the terms are +, the quotient must be + ; because in the divisor ×+ in the quotient, produces + in the dividend.

in the

2. When the terms are both - the quotient is also +; because in the divisor + in the quotient, produces dividend.

3. When one term is + and the other -, the quotient must be -; because in the divisor X in the quotient produces in the dividend, or in the divisor ×+ in the quotient gives in the dividend.

-

So that the rule is general; viz. that like signs give +, and unlike signs give

in the quotient.

CASE

CASE II.

When the Dividend is a Compound Quantity, and the Divisor a

Simple one:

DIVIDE every term of the dividend by the divisor, as in the former case.

4. Divide 6ab-8ax + a by 2a.

5. Divide 3x2— 15 + 6.x + 6a by 3x. 6. Divide 6abc + 12abx-9a'b by 3ab. 7. Divide 10a3x-15x2-25x by 5x. 8. Divide 15a2bc-15acx2 + 5ad2 by — 5ac, 9. Divide 15a + 3ay-18y2 by 21a. 10. Divide - 20d b2 + 60ab3 by - 6ab.

CASE III.

When the Divisor and Dividend are both Compound Quantities:

1. SET them down as in common division of numbers, the divisor before the dividend, with a small curved line between them, and ranging the terms according to the powers of some one of the letters in both, the higher powers before the lower.

2. Divide the first term of the dividend by the first term of the divisor, as in the first case, and set the result in the quotient.

3. Multiply the whole divisor by the term thus found, and subtract the result from the dividend.

4. To this remainder bring down as many terms of the dividend as are requisite for the next operation, dividing as before; and so on to the end, as in common arithmetic.

Note

Note. If the divisor be not exactly contained in the divi'dend, the quantity which remains after the operation is finished, may be placed over the divisor, like a vulgar fraction, and set down at the end of the quotient, as in common arithmetic.

a-c) a3-4a2c + 4ac2 —c3 ( a2-3ac + c2

a-2) a3-6a2 + 12a-8 (a2 - 4a + 4
a3-2a2

1. Divide a + 4ax + 4x2 by a + 2x.
2. Divide a3-3a2% + 3az2—z3 by a—z.

3. Divide 1 by 1 + a.

4. Divide 12x1- 192 by 3x-6.

Ans. a + 2x.

Ans. a2-2az + z2. Ans. 1-a + a2-a3 + &c.

Ans. 4x38x2 + 16x + 32.

5. Divide as-5ab10a3b2 - 10a2b3 +5ab4-bs by a2

2ab+b2.

Ans. a3-3ab + 3ab2 —b3.

6. Divide 48z3 — 96az2—64a2z + 150a3 by 2z-3a.

7. Divide b-3b2x2 + 3b2x2-x by b3-3b2x + 3bx2-x3.

8. Divide a1-x1 by a-x.

9. Divide a3 + 5a2x + 5ax2 + x3 by a + x.

10. Divide a + 4a2b2 - 3264 by a + 2b.

11. Divide 24a* — ba by 3a-2b.

ALGEBRAIC FRACTIONS.

ALGEBRAIC FRACTIONS have the same names and rules of operation, as numeral fractions in common arithmetic; as appears in the following Rules and Cases.

CASE