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x3-3x2y+3xy2—y3 8√✅ax+2√√x2y—4√] az—√√✅ xyz

1. Multiply -y by x+y.

2. Multiply x2+2xy+y2 by x+y.

Ans. x2-y

Ans. a+3xy+3.xy2+y3

3. Multiply x+xy by x—y.

Ans. x3+x2y3—x3y—xyŝ

4. Multiply/xy−√✅az by √✅✅ax+by.

Ans. ax2y-a2xz+√√/bxy2—√/ abyz.

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DIVISION of Algebraic quantities, as in common Arithmetic, is the reverse of Multiplication, and is commonly divided into three cases.

CASE I.

When the quantities are both simple.

RULE.-Divide the coefficient of the dividend, by the coefficient of the divisor, to obtain the coefficient of the quotient; expunge the letters which are common to both quantities, and place the remaining ones to the coefficient of the quotient.

If the signs of the dividend and divisor be like, the sign of the quotient will be +; but if the signs be unlike, the sign of the quotient will be minus.

In simple quantities, place the dividend above a small line, and the divisor under it, in the manner of a vulgar fraction.

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When the dividend is a compound quantity, and the divisor a simple one.

RULE.-Divide each term of the dividend by the divisor separately, and the resulting quantities, with their proper signs, will be the quotient required.

EXAMPLES.

1. Divide 2xy+4x2 by 2x.

2x)2xy+4x2

y+2x

2. Divide 8x2y-12xy+4x2z by 4xy.

4xy)8x2y-12xy+4x2z

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Such terms in the dividend, of which the divisor is not a common multiple, may be placed as a fraction, see example second.

If the dividend be a simple quantity, and the divisor a compound one, they may be placed as a vulgar fraction, see example fourth,

3. Divide 6x2y2-4x3y+8xy by —2xy.

-2xy)6x2y2-4x3y+8xy

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Ans. 2y+xy—5z.

2. Divide 12xy2- 6x2y2+30xyz by-6xy.

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5. Divide 9xy-12ax+14x2y2 by 4xy.

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When the divisor and dividend are both compound quantities.

RULE. Arrange both the divisor and dividend according to the powers of the same letters, beginning with the highest power; then find how often the first term of the divisor, is contained in the first term of the dividend, and place the result for a quotient, observing that if the first term in the divisor, and the first term in the dividend, have like signs, the term in the quotient will be plus, if unlike signs, minus.

Multiply each term in the divisor by this quantity, and place the product under the corresponding terms in the dividend; then proceed as in subtraction. To the remainder, bring down the next term, or as many terms as may be wanted.

If there be a remainder after the division, it must be placed in the quotient, in the form of a vulgar fraction, with its sign placed before it, as in examples 4th and 5th.

EXAMPLES.

1. Divide x3-3x2y+3xy2—y3 by x-y.
x—y)x3-3x2y+3xy2—y3(x2—2xy + y2
x3- x2y

-2x2y+3xy2 -2xy+2xy

xy2-y3 xy2 y3

In this example the terms are arranged according to the rule, beginning with x3.

1. The first term, x, of the divisor, and the first term, x3, of the dividend, have like signs, and x is contained in x3, x2 times; therefore, x2 will be the first term in the quotient.

2. Multiply x-y, by x2, and it gives x3—x2y.

3. Subtract 3-x2y, from x3-3x2y, and there remains -2x2y; to this remainder, bring down the next term +3xy2.

4. Here the first term of the dividend, -2a2y, is a minus quantity, and the first term, x, of the divisor, is plus, that is, they have unlike signs, and a being contained in -2x2y, -2xy times; therefore, the term in the quotient will be -2xy.

5. Multiply and subtract as before, and the remainder will be ay2; to this remainder bring down the next or last term y3.

6. being contained in xy2, +y2 times, put this term in the quotient and proceed as above; then the quotient will be found to be a2-2xy+y2.

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