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3. Multiply - 2xa + a2 by x2 + 2xa + a2.

4. Multiply 9-7x+3x-5 by 2x-5x+3, by detaching the coefficients.

5. Multiply 15a-13ax +19a2x2- 2x by 5a3 + 17a3x -15, by detaching the coefficients.

6. Multiply 4a + 3a3x + 5ax3 + 2x1 by 2a3 + 7ax2 + x3, by detaching the coefficients.

7. Multiply a + a2 by a3 - a, by detaching the coefficients.

8. Multiply 3x-5y by 8x3-13y, and the product by 1522 - 2y3.

9. Develop (92 - 2y3)3, or find the third power of 9x-2y3.

10. Find the ninth power of xy, or develop (x - y)o. 11. Multiply 3a23 + 5bx2 + 2cx+d by 4x2 + 3x + 5. 12. Multiply 7a2x2 + 9ax2 + 11x by 7a2x2 - 9ax + 11x. 13. Multiply a3 — a2x + ax2 — x3 by a +x, and the product by a +4.

14. Multiply 1 + (1 + a)x + (1 + a + b)x2 + (1 + a + b c) by 1-x.

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2-3, and the product

15. Multiply - 37x-22 by by2+7x, and the last product by x 6.

16. Develop (7 — 3x)2 . (4 — 5x)3 . (2 — 9x)*.

17. Multiply 4a2 − x + 4 by x + 1, and the product by x + 2, and the last product by a2 — 3.

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18. Find the product of 4x3 + 5x2a — 6xα2 — 7a3 and

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19. Develop (17a3b3c — 32pq) and (2x2 — 5yz)7.

20. Multiply aa2 + bx”yTM — cy3m by x" + dym.

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21. Multiply 2-3x + 3x2 - 3x3 + 3* -, etc., by 1 + ∞. 22. Multiply a + b + c by a-b-c, and develop

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23. Multiply an−1 + an-2f + an—8f3 + an−173 +

by a-b.

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24. Multiply a−9 by a +56 −6c+ 9, and de

velop (x + 2vwq)*.

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SECTION V.

DIVISION.

(1.) DIVISION may be considered as the reverse of Multiplication; that is, as having given a product and one of its factors, to find the remaining factor.

(2.) The product or quantity to be divided is called the dividend, and the factor by which it is to be divided is called the divisor. The remaining factor is called the quotient, because it denotes how often the dividend contains the divisor. If the divisor is contained in the dividend a certain number of times, and if there is any portion of the dividend left which does not exactly contain the divisor, the part left is called the remainder; if there is no remainder, the division is said to be exact.

(3.) It may be proper, before entering upon the rules of Division, to premise the following remarks.

By definition 8, each of the expressions, ab, a: b,

a

B'

is

used to signify that a is to be divided by b; also the same thing is sometimes denoted by ba. French authors frequently put the divisor to the right of the dividend, and having drawn a right line between the dividend and divisor, and another right line below the divisor, they put the quotient under the last line; thus, if a is the dividend, b the divisor,

b

and q the quotient, then a signifies that if we divide a by b, we shall have q for the quotient.

a

The form is called a fractional expression, a being called the numerator and 6 the denominator of the fraction; so that what is called the dividend in Division is the numerator of the fraction, and the divisor is called the denominator of the fraction; also a and b are sometimes called the terms of the fraction.

It is manifest that the divisor and dividend must both be quantities of the same kind, abstract numbers; or that the divisor must be an abstract number.

When a and b are quantities of the same kind, or abstract numbers, the fraction is often called the geometrical ratio

b

of a to b, or, more properly, it is said to represent the geometrical ratio of a to b.

Again, if a, b, c denote any three integral quantities, it is manifest, since ac is c times a, or c times as great as a, that

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a

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= : so that the b

fraction is multiplied by the integer c by multiplying its

b

numerator by c. Also, since be is c times as great as b, it is plain that is c times as great as or that <=c=

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a

a

a

bc'

α

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; hence, the fraction is divided by the integer c by

b

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a

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times, and contains c times, it

=

b

ac

be

b

be'

; so that the value of the frac

tion is not changed by multiplying its numerator and de

b

nominator by c.

Consequently, any integral factor, which is common to the numerator and denominator of any fraction, may be erased or stricken out of each without affecting the value of the fraction. Hence, also, any integral factor, which is common to the dividend and divisor in Division, may be stricken out of each without altering the quotient.

Let a, b, c, d denote any four integral quantities; then it is evident, since xc, and since c is d times as great

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tions are multiplied by each other, by taking the product of their numerators for a new numerator, and the product of their denominators for a new denominator.

с

Also, since c is d times as great as it is evident that

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a

bc

ad

bc

Hence, one fraction is divided by another by multiplying the numerator of the fraction to be divided by the denominator of the divisor, and the denominator of the fraction to be divided by the numerator of the divisor; or, which is the same thing, invert the divisor and proceed as in Multiplication.

From what has been proved, we have

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с

d

ac

= also bd'

b

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we get a ×

=

d d

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Consequently, the product of fractions whose numerators and denominators are integers, is independent of the order in which the fractions are written or taken in the multiplication; and the same is true of the product of an integer and a fraction, and indeed of the product of any number of integers by any number of fractions, as is evident from what has been done in Multiplication.

(4.) Again, since in Division the dividend is supposed to equal the product of the divisor and quotient, if the dividend and divisor are both positive, the quotient must clearly be positive; but if the divisor and dividend are both negative, the quotient must be positive, since the divisor and quotient must have unlike signs in order to give the sign to the dividend, which is their product, as appears from the rule of signs in Multiplication.

-

If the divisor has the sign+, and the dividend the sign -, it is manifest that the quotient must have the sign -, since the divisor and quotient must have unlike signs; also, if the divisor has the sign, and the dividend the sign +, the quotient must have the sign, since, by the rule of signs in Multiplication, the divisor and quotient must have like signs, in order that the dividend, which is their product, may have the sign+; consequently, since the divisor has the sign, the quotient must also have the sign

Hence we deduce the following rule for the sign of the quotient in Division.

RULE.

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If the divisor and dividend have like signs, the quotient must have the sign + ; but if the divisor and dividend have unlike signs, the quotient must have the sign Or we may express the rule more concisely, by saying that like signs give + and unlike signs give

(5.) Again, the same or different powers of any quantity are multiplied together by adding their exponents. Hence, since Division is the reverse of Multiplication, we must change the sign of the exponent of the quantity in the divisor (or conceive the sign to be changed), and then proceed as in Multiplication.

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Thus, if we have to multiply a by a we get a × a = and to divide a by a we have a × a-1 = a11 = ao, which

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tains itself once; consequently, any quantity which has 0 for an index, is an expression for 1. The expression shows that it has been obtained by dividing the quantity by itself. Again, to divide 6abc by 3ab, we have 6a2b3c ×

3-1a-b-5-4-2 × 3oa-4b-2-3

=

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6a2b3c 2
3ab

=

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so that 3o=

; consequently, any quantity

with a negative exponent is an expression for 1, divided by the quantity with a positive exponent; which is in conformity to the notation adopted in (11) of the definitions.

Hence, any quantity can be carried from the denominatorof a fraction into the numerator, by changing the sign of its exponent, so that any fractional quantity may be changed to an integral form; thus, for we may write ab-c-2.

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Reciprocally, any integral form may be reduced to a frac tional form; thus, abs may be expressed by

1

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so that

1 is the numerator of the fraction, and the product of the quantities, with the signs of their exponents changed, is thedenominator.

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