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of the result, without the necessity of performing an actual multiplication.

EXAMPLES. 1. (32+3)2 =

6. (5a2 +7ab)2= 2. (3a +36)2=

7. (503 +8a+b)2= 3. (5a +36)2=

8. (2a+1)= 4. (5a2 +25)=

9. (1+})= 5. (5a +b)=

10. (3+})=

67. The square of the difference of two numbers is equal to the square of the first, minus twice the product of the first by the second, plus the square of the second.

Thus, if we multiply a-6 by

a” - ab

ab+12 we obtain the product a- 2ab +62.


1. (20—30)2=
2. (5a-40)=
3. (6a - x)?=
4. (6a2 – 3x)=
5. (x-*y)=


6. (702–12ab)2=
7. (7a2b2 12ab)2=
8. (243–5)=
9. (2-})?=
10. (4-})=

68. Meaning of the sign E. Since

(a +-7)=a2+2ab +7?, and

(ab)?=~2—2ab+82, we may write both formulas in the following abbreviated form,

(a=0)2=a2+2ab +32; which indicates that, if we use the + sign of b in the root, we must use the + sign of 2ab in the square; but if we use the

sign of b in the root, we must use the sign of 2ab in the square. By this notation we are enabled to express two distinct theorems by one formula.

69. The product of the sum and difference of two numbers is equal to the difference of their squares. Thus, if we multiply

a+b by

a2 + ab

-ab-62 we obtain the product

a? -62.



1. (3a+26) (30—26)=
2. (7ab+x) (7abx)=
3. (8a+7bc) (82—7bc)=
4. (5a2 +663) (522—663)=
5. (4a2 +3mx) (4a2—3mx)=
6. (3a2b+) (3a2b-a)=
7. (m+1) (m-1)=

8. (4+5) (4-3)= The pupil should be drilled upon examples like the preceding until he can produce the results mentally with as great facility as he could read them if exhibited upon paper, and without committing the common mistake of making the square of a+b equal to a2 +22, or the square of a-6 equal to aż-62.

The utility of these theorems will be the more apparent when they are applied to very complicated expressions. Frequent examples of their application will be seen hereafter.


DIVISION. 70. Division is the converse of multiplication. In multiplication we determine the product arising from two given factors. In division we have the product and one of the factors given, and we are required to determine the other factor.

The dividend is the product of the divisor and quotient, the divisor is the given factor, and the quotient is the factor required to be found.

When the divisor and dividend are both monomials.

71. Since the product of the numbers denoted by a and b is denoted by ab, the quotient of ab divided by a is b; that is, ab-a=b. Similarly, we have abc-a=bc, abc--c=ab, abc---ab =c, etc. The division is more commonly denoted thus:


- ac, =ab,




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So, also, 12mn divided by 3m gives 4n; for 3m multiplied by 4n makes 12mn.

72. Rule of Exponents in Division.-Suppose we have as to be divided by a?. We must find a quantity which, multiplied by a?, will produce as. We perceive that a’ is such a quantity; for, according to Art. 58, in order to multiply a3 by aż, we. add the exponents 2 and 3, making 5; that is, the exponent 3 of the quotient is found by subtracting 2, the exponent of the divisor, from 5, the exponent of the dividend.

Hence, in order to divide one power of any quantity by another power of the same quantity, subtract the exponent of the divisor from the exponent of the dividend.

73. Proper Sign of the Quotient.— The proper sign to be prefixed to a quotient may be deduced from the principles already established for multiplication. The product of the divisor and quotient must be equal to the dividend. Hence, because tax(+6)=+ab,

tab:(+)=ta. -ax(+b)=-ab,


-ab-(+6)=-a. tax(-5)=-ab,

-ab:-(-6)=+a. -ax(-6)=+ab,

+ab(-6)=-a. Hence, if the dividend and divisor have like signs, the quotient will be positive; but if they have unlike signs, the quotient will be negative.

74. Hence, for dividing one monomial by another, we have the following


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

2. To this result annex all the letters of the dividend, giving to each an exponent equal to the excess of its exponent in the dividend above that in the divisor.

3. If the dividend and divisor have like signs, prefix the plus sign to the quotient; but if they have unlike signs, prefix the minus sign.

EXAMPLES. 1. Divide 20ax3 by 4x.

Ans. 5ax. 2. Divide 25aRxy4 by –5aya.

Ans. -5a-ay. 3. Divide – 72ab6x2 by 1263x.

Ans. -6ab?x. 4. Divide 77a3b5c6 by – 11ab3c4.

Ans. 7a2b2c2. 5. Divide 48a3b3c4d by - 12abc. 6. Divide - 150a5b8cd3 by 30a%b4d2. 7. Divide - 250a%b8x3 by -5abx3. 8. Divide 272a3b4cxc6 by – 17a2b3cac. 9. Divide – 42a6b3c by 21abc. 10. Divide - 300a3b4c by -50bx.

75. Value of the Symbol a'.—The rule given in Art. 72 conducts us in some cases to an expression of the form a°. Let it be required to divide az by a?. According to the rule, the quotient will be a2-2, or a'. Now every number is contained in itself once; hence the value of the quotient must be unity; that is,

a'=1. To demonstrate this principle generally, let a represent any quantity, and m the exponent of any power whatever. Then, by the rule of division,

am-am=am-m=a'. But the quotient obtained by dividing any quantity by itself is unity; that is, a=1, or any quantity having a cipher for its exponent is equal to unity.

76. Signification of Negative Exponents.—The rule given in Art. 72 conducts us in some cases to negative exponents. Thus, let it be required to divide a3 by a'. We are directed to subtract the exponent of the divisor from the exponent of the dividend. We thus obtain

a3-5, or a-2

a3 But a: divided by ao may be written az; and, since the value of a fraction is not altered by dividing both numerator and denominator by the same quantity, this expression is

1 equivalent to al'

1 Hence a-% is equivalent to

a So, also, if a’ is to be divided by a®, this may be written

a? 1

@5=az=a In the same manner, we find



am that is, any quantity having a negative exponent is equal to the reciprocal of that quantity with an equal positive exponent.

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