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Examples.

Ex. 1. (2a-3b)'=4a-12ab+9b".
Ex. 2. (5ab-2x)'=25a'b'-20abx+4x2.
Ex. 3. (8a-3x)'=64a-48a2x+9x2.
Ex. 4. (6a2-2b)'=36a*—24a3b+4b3.

Ex. 5. (7a2-10ab)2=49a*-140a3b+100a2b2.
Ex. 6. (7a'b'-12ab)'=49a'b'-168a b'+144a3b'.
Here, also, beginners often commit the mistake of
putting the square of a-b equal to a2-b'.

THEOREM III.

(67.) The product of the sum and difference of two quantities is equal to the difference of their squares. Thus, if we multiply a +b

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Ex. 1. (3a+4b) (3a-4b)=9a-16b3.
Ex. 2. (6ab+3x)(6ab-3x)=36a2b2—9x2.
Ex. 3. (7a+2b)(7a-2b)=49a-4b'.
Ex. 4. (8a+7bc)(8a—7bc)=64a2-49b2c2.
Ex. 5. (5a2+6b3) (5a3—6b3)=25a*—36bo.

Ex. 6. (5x3y+3xy2)(5x3y—3xy3)=25x*y2—9x'y'. The student 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.

QUEST.-Illustrate by examples. What is the product of the sum and difference of two quantities? Illustrate by examples.

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

(68.) The same theorems will enable us to resolve many complicated expressions into their factors.

Ex. 1. Resolve a'+2ab+b2 into its factors.

Ans. (a+b)(a+b).

Ex. 2. Resolve n'+2n+1 into its factors.

Ans. (n+1)(n+1).

Ex. 3. Resolve a2-2ab+b2 into its factors.

Ans. (a—b)(a—b).

Ex. 4. Resolve a'-6ab+962 into its factors.

Ans. (a-3b)(a-3b).

Ex. 5. Resolve a-b' into its factors.

Ans. (a+b)(a−b).

Ex. 6. Resolve a-b' into its factors.

Ans. (a2+b2)(a'—b3).

QUEST.-How may complicated expressions be resolved into factors?

SECTION V.

DIVISION.

(69.) Division consists in finding how many times one quantity is contained in another. The quantity to be divided is called the dividend; and the quotient shows how many times the divisor is contained in the dividend.

When we have obtained the quotient, we may verify the result by multiplying the divisor by the quotient— the product should be equal to the dividend. Hence we may regard the dividend as the product of two factors, viz., the divisor and quotient-of which one is given, that is, the divisor; and it is required to find the other factor, which we call the quotient.

CASE I.

(70.) When the divisor and dividend are both moomials.

Suppose we have 72 to be divided by 8. We must find such a factor as multiplied by 8 will give exactly 72. We perceive that 9 is such a number, and therefore 9 is the quotient obtained when we divide 72 by 8.

Also, if we have ab to be divided by a, it is evident that the quotient will be b; for a multiplied by b gives the dividend ab.

QUEST.-What is the object of division? What is the divisor? What is the quotient?

What is the dividend?
What is case first?

If we divide 60x by 5, we obtain 12x, for 12x multiplied by 5 gives 60x.

So, also, 12mn divided by 3m gives 4n.

Again, suppose we have a to be divided by a'. We must find a number which, multiplied by a2, will produce a3. We perceive that a3 is such a number; for, according to Art. 56, we multiply a3 by a2 by adding 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 we derive the following

RULE OF EXPONENTS IN DIVISION.

(71.) A power is divided by another power of the same root, by subtracting the exponent of the divisor from that of the dividend.

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(72.) Let it be required to divide 48a" by 6a2. must find a quantity which, multiplied by 6a2, will produce 48a. Such a quantity is Sa3; for, according to Arts. 55 and 56, 8a3×6a' is equal to 48a". There

QUEST.-How are powers divided?

fore, 48a divided by 6a2 gives for a quotient 8a'; that is, we have divided 48, the coefficient of the dividend, by 6, the coefficient of the divisor, and have subtracted the exponent of the divisor from the exponent of the dividend.

Hence, for the division of monomials, we have the following

RULE.

1. Divide the coefficient of the dividend by the coefficient of the divisor.

2: Subtract the exponent of each letter in the divisor from the exponent of the same letter in the div idend.

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QUEST. Give the rule for the division of monomials.

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