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10a3 +11ba2+362 Ja5b2c+15bc2 by 5a23ba-5bc. -15c19bc

10a3+11b a2+3b2 a-5b2c+15bc2 ( 5a2+3ba-5bc

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11. Divide 12a3b2 29a3bc +15a3c2 +23a2b3 —— 31a2b3c -9a2bc2 + 15a2c3 +10ab1+6ab2c2 by 3ab 5ac + 2b2.

Dividend arranged

12ba3+23b3 \a2 + 10b1a

31b2c

6b2c2

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Product of the divisor by the first term of the quotient.

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In this example, when we wish to find the quotient of the first term of the dividend by the first term of the divisor, we first consider that a3 divided by a, gives a2; we then divide the quantities which stand at the left of the vertical line in the dividend, and are understood to be multiplied by a3, by the quantities multiplied into a in the divisor; the quotient is that part of the general quotient which is multiplied by a2. The partial quotient so obtained is then multiplied into the whole of the divisor, and the product subtracted from the whole dividend.

3

In the preceding example, we take the quantity 1263—29bc +15c2, which is multiplied by a3, and divide it by 36-5c, which in the divisor is multiplied by a.

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Performing the operation, we have 4b-3c; and since a3, divided by a, gives a2, this quantity is to be multiplied by

a2; and hence the first term of the quotient is 4b|a2.

-3d

After the multiplication of the divisor by this term, and the subtraction of the product from the dividend, we have, for the first term of the new dividend, certain quantities multiplied into a2; a2 divided by a gives a; and dividing the quantities multiplied by a2 by those multiplied by a, we have,

1563-25b2c 9bc2+15c3 3b-5c

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562 3c2

a is therefore the second term of the quotient. Multiplying this into the whole divisor, and subtracting the product from the second dividend, 0 remains; and hence the division is completed.

12. Divide 12ab26a b2 + 10ab3 + 18a3b3-19ab - 30a3b1 +25a3b2 +8a3 +9a2b3· +6ab2 by 3ab5a2b2+20.

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10a2

15a2b+12ab2·

Divisor arranged.

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Quo. b33b2x + 3bx2 + x3.

Quo. a2+4ax + x2.

15. Divide b3-3b1x2+3b2x2—x by b3—3b2x+3bx2—x3.

16. Divide a3 + 5a2x + 5ax2+x3 by a + x.

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20. Divide a +8a3x + 24a2x2 + 32ax3 +16x1 by a +2x.

Quo. a3+6a2x + 12ax2 + 8x3.

21. Divide a x + x2-x by x2-x.

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Quo. x2-x+1.

(17.) We shall now proceed to show some of the applications of algebra, in investigating the properties of numbers. 1. Suppose we have two numbers, the sum of which is 48, and their difference 10, and it is required to find those numbers.

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