Note III. Operations in division may often be facilitated by the formulæ given in the last note to multiplication, as well as by the following.

Remember that 2n denotes an even exponent, 2n + 1 an odd exponent; then b2" is divisible by a − b, by a + b, or by a2

Thus,

2n

b, nor by a + b.

- b.

b, but not by a + b. + b, but not by a —

ab + ab2 + b3,

a2b + ab2 — b3,

a2 + a3b + a2b2

b.

+ ab3 + b1,

b2.

(a + b3) ÷ (a + b)

(a3 — b3) ÷ (a + b) = a1 — a3b + a2b2

where the latter is evidently not a complete quotient.

265

· ab3 + b4

a+b'

These theorems will enable the student to effect important simplifications in the reduction of fractions, and of equations, and must therefore obtain sufficient attention before he proceeds further.

DIVISION BY DETACHED COEFFICIENTS.

As it was shown, in the note on multiplication, that the multiplication may be very conveniently carried on by means of the detached coefficients only, so it may be readily shown that the same can be done in division; and its practice is earnestly inculcated on the student for precisely the same reason as it was there done, its economy of time and space, and especially as an introduction to the recent improvements made in the solution of numerical equations. Thus, for example, to divide 3x3 bx by x3- bx -C.

1 + 0 − b − c) 3 + 0 − b + 0 (3 + 0 + 26 + 3c + 262 + 5bc...

And since the highest power of x is

203

o, we have for the result 30+ Ox¬1

+2bx2 + 3 cx3 + 262x2+5bcx5 + ad inf.

....

This example has been adopted on account of its containing both literal and zero, as well as numeral, coefficients.

SYNTHETIC DIVISION.

WHEN one algebraic function of a quantity is to be divided by another, the

coefficients of each being given in numbers, the following process, invented by the late Mr. Horner to subserve the solution of numerical equations, is of the utmost value.

1. Write down the coefficients of the dividend in a horizontal line with their proper signs, and where a term is wanting write O in the place of its coefficient. 2. Draw a vertical line before the first term, and to the left of this line put down the coefficients of the divisor, with the same precaution respecting absent terms, but the signs of these coefficients changed; and having them so disposed that the first coefficient is in a line with the horizontal column spoken of in (1). 3. Bring down the first coefficient of the dividend: this will be the first term of the quotient.

4. To obtain the others in succession, multiply the immediately preceding term of the quotient by the remaining terms of the divisor, having their signs changed; and place them successively under the corresponding terms of the dividend in a diagonal column, beginning at the upper line. Add the results in the second column, which will give the second term of the quotient; and multiply the terms of x in the divisor by this result, placing the products in a diagonal series, as before. Add the next series of results, which will give the next coefficient of the dividend; and multiply x by this again, placing the products as before. This process, persevered in till the results become 0, or till the quotient is determined as far as necessary, will give the same series of terms as the common mode of division, or as the division by detached coefficients, in the last article, when carried to an equivalent extent.

Let us take as an example the division of x6 - 6x520x 40x + 100 by x3 2x2+5x

40x3 +50x2 - 9. Following the prescribed directions with respect to arrangement, we have the horizontal and vertical columns at once.

-

Multiply each term of the quotient in succession by all the terms of the divisor, (the first or 1 excepted, the upper line standing for the result of that step,) carrying the results to the places denoted by the corresponding powers of the quantity. This will always be done when the deficient terms are supplied by zero, to preserve the places as in arithmetic, by carrying them out diagonally to the right, or moving one step to the right in making the commencement of each successive row. Thus we obtain the diagonal series 1 + 2 -5+9. Add the vertical column — 6 + 2, and with the result 4, multiply all the terms of the divisor as before, giving the next diagonal series 8+ 20 36. Add the third column, and obtain the result + 7; and by this obtain another diagonal column + 14 3563, and then another sum + 3. Proceed in the same manner till the results either terminate in zeros, or have been carried far enough to answer the purpose in view. In the above work nine terms are obtained to which the powers of x (the highest being a6-3 = x3) may be attached as they stand, and the quotient is a3 15x-1 4x2+7x+3 +158x3+291x-4

406x-5

......

ad infinitum.

22x-2

It is to be understood that the coefficient of the leading term of the divisor is 1; and in cases where this does not occur, it can be made so, by dividing every coefficient of the divisor and dividend by that coefficient.

With the view of illustrating the operation, it will be advisable to work the same question in the usual way, employing, however, only the detached coefficients.

1−2+5—9) 1−6+20—40+50—40+100 (1—4+7+3−15-22+158+291-406

The connexion between this and the synthetic division will best appear by taking a form intermediate between the two: viz. by placing the subtrahends in order, having their signs changed, but still in the horizontal position which they occupy in the old method.

Divisor.

Dividend.

Quotient.

1—2+5—9) 1-6 +20-40 +50-40 +100 (1-4+7+3—15—22+158 +291-406

The relation of this to the common method is obvious.

Had we, however, left out the numbers marked with the asterisk in this work, the sums would severally have been the terms of the divisor; and hence, if we omit multiplying by — 1 (the first coefficient of the divisor with its sign changed) the line now marked as "remainder" might have been employed for the terms

VOL. I.

K

of the quotient, which are the sums of the several columns. This is in accordance with the rule, which requires the first coefficient 1 to be omitted; and the change in the signs of all the other terms is effected by changing the remaining signs of the divisor before we begin to operate.

Further, to avoid bringing the work so far down the page, leaving so much space unoccupied on each side of the diagonal columns, the several products of the coefficients of the modified divisor by the successive quotient figures, may be themselves set down in diagonal columns: thus, instead of

In comparing this mode of working with the preceding, we remark that: 1st. The coefficients which appear as subtrahends in the old method, appear as addends having their signs changed, in the new. The change of all the signs of the divisor except the first, in the new method secures this.

2nd. No coefficient is used till we arrive at the vertical column in which it appears, and which occurs immediately after that column is completed. This arises from only completing at each step the first term of what constitutes the remainder in the old method.

3rd. The work is contracted into a series of horizontal columns, in number equal to the terms of the divisor, without descending the page continually, as in the old method. This is effected by carrying the first term of each product to the upper line, and gradually descending in a diagonal line with the others.

4th. The work besides not descending on the page, does not extend across it so far as in the old method. This arises from the less breadth occupied by the divisor in its vertical than in its horizontal arrangement; and from the quotient falling beneath the work instead of being placed to the right, as in the ordinary method.

This last process completes the Algorithm of the method, and brings us to the rule as above laid down in every particular.

On the ground of economy of time alone, this method does not require half so much writing as the ordinary one; and the chances of mistake in the operation are lessened in a still greater degree *.

* In the example just given, if we compare any two corresponding columns, as that belonging to a4 for instance in the two methods, they will stand thus:

If again we estimate the total saving, beside the compactness of its disposition and the fewer chances of error, we shall discover that

x occurs 67 times more in the old than in the new process;

[Indices

EXAMPLES.

1. Divide x - y6 by x — y; or, which is the same thing,

x2 y2 z2 + x3 y3 z3

1 xyz

into an infinite series; and likewise

(or) into a series, and show that is the symbol of an infinitely great

quantity.

PROBLEMS AND THEOREMS ON THE FIRST FOUR RULES OF

ALGEBRA.

1. HALF the difference of two quantities added to half the sum gives the greater of them, and subtracted leaves the less.

[Let the student select his own symbols, and illustrate it with his own numbers.]

2. If 2s = a + b + c, what are the values of s a, s - b, and s c? and what is half their sum equal to ? Find also their product, and arrange its terms systematically.

3. The difference between the square of the sum of two numbers and the square of their difference is equal to four times their product; and the sum of the squares of their sum and difference is double the sum of their squares. Prove this.

4. The sum of two numbers multiplied by their difference is equal to

Indices without sign, (or + understood,) 32 times more;

Indices with sign 25 times more;

......

to which the system of detached coefficients has the advantage in common with the synthetic division.

In the detached system there occur 172 figures besides the answer; in the new only 72, or 100 less; and of the signs + and —, the number in the detached operation is 66, and in the other 32, or rather less than half the number.