NOTE. — If the dividend is not exactly divisible by the divisor, the remainder must be placed over the divisor in the form of a fraction and connected with the quotient by the proper sign. 2. Divide + 4y by x2-2xy + 2y3. x2-2xy +2 y3) x* + 4 y* x-2xy + 2x2 y2 (x2+2xy + 2 y2 2x3y-2xy2+4y1 2 x3y - 4x2 y2+4 x y3 2 x2 y2-4 x y + 4 y1 2 x2 y24 x y3 + 4 y1 NOTE. By multiplying the quotient and divisor together all the terms which appear in the process of dividing will be found in the partial products. 3. Divide 24-1 by x-1. x − 1) x1 — 1 (×3 + x2+x+1 4. Divide a xay + bx-by + z by x-y. x − y) a x − a y + b x − b y + z (a + b + x = y 5. Divide 2by2by-3 by z + 6 b3y + by zyz by 2b-z. Ans. 3by-by+y. 6. Divide +23 by c+x. Ans. ccxx3. 7. Divide a3+ a2 + a2 x + a x + 3 ac + 3c by a + 1. Ans. a+ax +3 c. 8. Divide a b-d-ax-bx+ dx by a+b-d. 9. Divide 2 a 13 a3 y + 11 ay - 8ay + 2y by 2 a2-a y + y2. Ans. a6 ay + 2y2. 10. Divide a3-3 ab+3ab2b3 by a2-2ab+b2. 11. Divide 2 x3-19x2+26 x 19 by x 8. 20. Divide a a2x2 + 2 a x23 — x1 by a2 — a x + x2. 21. Divide x 2 x3 y3 + yo by x2 — x y + y2. 22. Divide 1-a by 1+a+a2 + a3. Ans. 1-a. 23. Divide 10 x3-20 x y + 30 y3 by x+y. 24. Divide 7 a x2 + 21 a x3 + 14 a by x + 1. Ans. 7 a 14 a x2-14 a x + 14 a. 25. Divide 27 a3 y 8 ay by 3y2-2ay. SECTION VIII. DEMONSTRATION OF THEOREMS. 57. FROM the principles already established we are prepared to demonstrate the following theorems. THEOREM I. The sum of two quantities plus their difference is twice the greater; and the sum of two quantities minus their difference is twice the less. Let a and b represent the two quantities, and a >b; their sum is a+b; their difference, a b. PROOF. 1st. (a+b)+(a−b) = a + b + a−b=2a; Therefore, when the sum and difference of two quantities are given to find the quantities, RULE. Subtract the difference from the sum, and divide the remainder by two, and we shall have the less; the less plus the difference will be the greater. In the following examples the sum and difference are given and the quantities required. THEOREM II. 58. The square of the sum of two quantities is equal to the square of the first, plus twice the product of the two, plus the square of the second. Let a and b represent the two quantities; their sum will be a+b. 59. The square of the difference of two quantities is equal to the square of the first, minus twice the product of the two, plus the square of the second. Let a and b represent the two quantities, and a > b; their dif ference will be a - b. 60. The product of the sum and difference of two quantities is equal to the difference of their squares. Let ab be the sum, and a b the difference of the two quantities a and b. 61. This theorem suggests an easy method of squaring numbers. For, since a2= (a - b) (a + b) + b2, 992 (991) (99 + 1) + 1298 × 100+1 = 9801. = 4972494 X 500 + 9 = 247 × 1000 + 9 = 247009. |