a-am-16 { a-b am-1 am-16 bm am-bam-2Z2 +a-2b+a-3b2+a-4b3+ ab + ab 10a3+ 11ba2 + 3b2a-56°c + 19 15bc2 by 5a2 +3ba-5bc. 10a2+11b|a2+382a5bc+15bc2 $5a2+3ba5bc + b-3c 11* 11. Divide 12a3b2 - 29a2bc +15a3c2 + 23a2b3-31a2bc 9a2bc2 + 15a2c3 + 10ab46abc2 by 3ab - 5ас + 262. Dividend arranged 126 a3+2363 a2 + 10b4a 29bc +15c Product of the divisor by the first ( term of the quotient. 15b3a2 + 106 a 25b2c-662c 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. 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. Performing the operation, we have 4b-3c; and since a3, divided by a, gives a2, this quantity is to be multiplied by 4ba2. a; and hence the first term of the quotient is -30 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 a by those multiplied by a, we have, 562 302 a is therefore the second term of the quotient. Mul tiplying this into the whole divisor, and subtracting the product from the second dividend, 0 remains; and hence the division is completed. 12. Divide 12ab 26a2b2 + 10a4b3 + 18a3b3 - 19ab -30a2b+25a2b2 + 8a3 +9a2b3 - 15a2b+ + 12a2b2 +6ab2 by 3a2b - 5a2b2 + 21 10a2 -26-5362 13. Divide 12x4 - 192 by 3x -6. Quo. 4x3+8x2+16x +32. 14. Divide a5 - 5a2b+ 10a3b2 - 10a2b+5abb5 by a2-2ab+b2. Quo, a3-3a2b+3ab2 - Б3. 15. Divide bo_3b4x2+3b2x2-x by b3-3b2x+3bx2-x3. Quo. b3 + 3b2x + 3bx2 + x3. Quo. a2+4ax + x2. 16. Divide a3 + 5a2x + 5ах2 + x3 by а + х. Quo. 1-a + a2 - a3+ a2 - &c. 19. Divide 25x5 - x2-2x3 - 8x2 by 5x3 - 4x2. Quo. 5x3+4x2+3x+2. 20. Divide a + Sax +24a2x2 + 32ax3 +16x4 by a+2x. Quo. a3+ 6a2x + 12ax2 + 8x3. 21. Divide 2 - x + x2-x by x2 -x. 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. |