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2. Divide a2+2xy + y2 by x+y.
x+y)x2+2xy + y2(x+y,
x2+ xy

xy+ y2 xy+ y2

3. Divide 62+13xy+6y2 by 2x+3y. 2x+3y)6x2+13xy+6y2(3x+2y 6.x2+9xy

4xy+6y2

4xy+6y2

4. Divide x3-9x2+27x-30 by x-3.

x—3)x3—9x2+27x—30(x2-6x+9—

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In the fourth question, -3 remains, hence, according to note at end of the rule, it is placed in the quotient in the form of a fraction, with its proper sign.

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7. Divide x3+x2y2+y3 by x2+x3yễ+;

'+x+y++ y2

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EXAMPLES FOR PRACTICE.

1. Divlde -2xy+y2 by x-y.
2. Divide x3-3x2y+3xy2+y3 by x+y.

Ans. x-y.

Ans. 2+2xy+y3.

3. Divide 6x4-96 by 3x-6.

4. Divide 1 by 1-a.

Ans. 2x+4x2+8x+16. Ans. 1+a+a2+a3+a1, &c.

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INVOLUTION is the raising of powers, or the method of finding the square, cube, biquadrate, fifth power, &c. of any given quantity.

RULE.-Multiply the given quantity into itself, as many times as there are units in the power, less one, and the last product will be the power required.

If the sign of the quantity to be raised be +, all its powers will be+; if, the powers will be + and alternately.

The square of x, is +x x+a=x2; the square of 2x2 is +2x2x+2x2 or 44; the square of -2r is -2xX-2x-4x2; (by note third to Case I. in multiplication) the cube of -2x is - 2x ×—2x ×-2x=4x2×-2x or -8x3.

EXAMPLES.

Suppose a quantity or root to be involved.

Then x2 will be its square,

x3

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cube,

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5th power,

6th

power, &c.

Also, suppose -2x, a quantity to be involved.

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Suppose a+y, to be the root or quantity to be

involved.

a + y a + y

a2+ay

ay + y2

a2+2ay+y2 the square,

a + y

a3+2a3y+ay2

a2y+2ay2+y3

a2+3a2y+3ay2+y3 the cube,

a + y

a++3a3y+3a2y2+ay3

a3y+3a2y2+3ay3+y+...

a1+4a3y+6a2y2+4ay3+y1 the fourth power, &c.

EXAMPLES FOR PRACTICE.

1. Required the square of 3x.

Ans. 9x2.

2. Required the cube or third power of 4x2.

3. Required the fourth power of 2ax2.

4. Required the cube of -3a2.

Ans. 64x6.

Ans. 16a1x3. Ans. -27a6.

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7. Required the fifth power of a+x.

8. Required the fifth power of a—x.

SIR ISAAC NEWTON's method of raising a binomial or residual quantity to any power whatever.

EXAMPLES.

1. Required the fourth power of a+x.

Here a+x+a+4a3x+6a2x2+4ax3+x1.

1. In raising a+ to the fourth power, a, which is the leading quantity, will have 4 for its index, that is, it will be at; this will be the first term.

2. The index, 4, will be the coefficient of the second term ; the index of the leading quantity a, will decrease 1 in each succeeding term; and the second quantity x, will come into the second term, and its index will increase 1, in each succeeding term throughout the operation; therefore, the second term will be 4a3r.

3. The terms already found are a1+4a3x.

4. Multiply 4, the coefficient of the second term, by 3, the index of a; divide the product 12 by 2, the number of terms already found, and the quotient 6 will be the coefficient of the third term.

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