(3.) 1—2a+3a2 —4a3 by 1+a. Ans, 1-a+a2—a3—4a1. (4.) a3-3a2x+3ax2-x5 by a-x. a4-4a3x+6a2x2-4ax3 +x4. a4 — x4. (5.) a3-a2x+ax2-x3 by a +x. (9.) (x+1)(x-2) (x−3) (x+4). x4-15x2+10x+24. (10.) (x−3) (x+4) (x−5) (x+6). x42x3-41x2-42x+360. (11.) (a+2x) (2a—x) (2a—3x) (3a+2x). (12.) (a-2x). a5-10a4x+40a3x2-80a2x3 +80ax1-32x5. 12a48a3x-39a2x2-8ax3 +12x1. DIVISION BY DETACHED COEFFICIENTS. 25. The same method may also be employed with great advantage in many cases in Division, wherever the divisor and dividend are such that the literal part of the terms of the quotient can be supplied by inspection. This method will be understood from the following examples :-Divide 3x4+14x3+12x+9 by x2+5x+1. Here placing the coefficients as in the margin, those of the divisor being to the right hand, immediately above those of the quotient, we find in the usual way, 3 for the first figure of the quotient; then multiplying the terms of the divisor by 3, the products are placed respectively under the three first terms of the dividend, and DIVISION BY DETACHED COEFFICIENTS. 13 the work proceeds exactly as if the letters were annexed; and since x2, the literal part of the first term of the x2 quotient is x2, and the powers will descend in the same or der as in the divisor and dividend, the quotient therefore is The process may be still farther abridged, as follows:Arrange the coeffi cients as before, and find the first figure of the quotient, which is 3; multiply the terms of the 3+140+ 12 +9 -3-15. -3 1 + 5+1 divisor by 3, and set 3+7 down the products under the dividend as before, but with their signs changed; then 14-15 — 1, which divided by 1, the first term of the divisor, gives 1 for the next figure of the quotient; multiply, and set down the products as before, with their signs changed; then 0-3+5=2, which again divided by 1, the first term of the divisor, gives 2 for the next figure of the quotient; multiply as before, and set down the products with their signs changed. Then supplying the letters, the quotient is 3x+2 as before, with the remainder 3x+7 2 + 5x+1 A little consideration will show that the two processes are identical, the several subtractions in the first being omitted in the second, and the signs of the partial products being conceived to be changed in the first, while they are actually changed in the second. When the divisor consists of two terms, and the coefficient of the first is unity, the division may be performed as follows: To divide 42433x2+8x +3 by + 3. Arrange the coefficients as in the margin, omitting the first term of the divisor; then multiply the first term 4 of the dividend by the 3 of the divisor; set down the product 12 under the second term, and subtract; the remainder is-12; 4 + 0. ·33 +8 + 3 (+3 1212 + 36 +9 - 3 6 Ans. 4x-12x2+3x x+3 which write down below; multiply the 12 by the 3 of the divisor, set down the product-36 under the next term, and subtract; the remainder is +3, which again write down below; proceed in like manner to the end, multiplying each remainder by the 3 of the divisor, and subtracting the product from the next term. In this case, the coefficient of the first term of the divisor being 1, to divide by it produces no alteration, and hence the first coefficient of the dividend, with the several remainders except the last, may be taken for the coefficients of the quotient. Ans. 2x-5. a-b. (1.) 4x2 —4x—15 by 2x+3. (2.) a2—2ab+b2 by a−b. (3.) 2x1—x3+x2+7x+3 by x2 −2x+3. 2x2 +3x+1. (4.) 12x1—x3-4x2 +7x-2 by 3x2 +2x−1. 4x2—3x+2. (5.) x4+4x+6x2+4x+1 by x+1. x2+3x2+3x+1. (6.) 3x110x+11x2-8x+6 by x-2. (7.) 3x1-4ax3 — 12a2x2 —7 a3x+a1 by X- -3a. (8.) 4a5+14a1x-5a3x2+8a2x3-14ax1+8x5 by a+4x. 4a4-2a3x+3a2x2-4ax3 + 2x1. (9.) 12a1-a3x-21a2x2+22ax3—6x1 by 3a2-4ax+2x2. 4a2+5ax-3x2. (10.) 6x5x4y+7x3 y2+3x2y3 +8xy+6y5 by 2x2 (12.) a5-32x5 by a-2x. a1+2a3x+4a2x2 + 8ax3 +16x1. 1−x+x2. (14.) 15a4+10a3x+4a2x2+6ax3-3x1 by 3a2x2+2ax. 5a2+3x2. MISCELLANEOUS EXERCISES. 1. Express algebraically the sum and difference of two numbers; then, first, add the two expressions; second, subtract the less from the greater; and, third, multiply the one : by the other what are the three results expressed in English? Ans. Twice the greater; twice the less; and the difference of the squares. 2. The sum of two numbers is 20, and their difference 12; find the numbers. Ans. 16 and 4. 3. Half the sum of two numbers is 15, and half their difference 8; find the numbers. Ans. 23 and 7, 4. If A can do a piece of work in 12 days, express the parts done in 1 day, and also in a days. Ans. and 12' 5. If be the less of two numbers, and 8 the difference, express the greater. Ans. +8. 6. If a be the greater of two numbers, and 8 the difference, express the less. Ans. x- 8. 7. Express the square of the sum of two quantities, a and b, and also the sum of their squares; and find the difference. Ans. a2+2ab+b2; a2+b2; 2ab. 8. If x represent a number of shillings, express the same sum of money in half-crowns. Ans. half-crowns. 2x 5 9. One of two brothers is as much above 25 years of age as the other is below it; if 25+≈ is the age of the one, what is the age of the other?" Ans. 25-x. 10. If x articles cost y pounds, find the price of each. Ans. y x 11. If one of two numbers in the ratio of m to n be expressed by mx, find the expression for the other. Ans. nx. 12. A sum of money is to be divided among three persons, A, B, and C; if x represent A's share, find B's and C's, when B's is double, and C's treble of A's; also when B's is half, and C's the third part of A's. Ans. 2x and 32; x and 2 x 3 B in b 13. If A can do a piece of work in a days, and days, find the part done by each in a days. Ans. and I a x 14. Express the two numbers whose digits are x and y; and x, y, and z. Ans. 10x+y; and 100x+10y+~. 15. What does x2 + y2 want of being a complete square? Ans. 2xy. 16 CHAPTER III. FRACTIONS. FUNDAMENTAL PRINCIPLES. 26. If the numerator and denominator of a fraction be both multiplied by the same number, or both divided by the same number, the value of the fraction is not changed. 27. A fraction is multiplied by any quantity, either by multiplying the numerator or dividing the denominator by that quantity. 28. A fraction is divided by any quantity, either by dividing the numerator, or multiplying the denominator by that quantity. 29. It follows from the first of these principles that a fraction may be reduced to its lowest terms by dividing its numerator and denominator by any factors which are common to both. In many cases such factors are easily discovered by inspection, and their product is called the Greatest Common Measure of the two quantities. 30. To find the Greatest Common Measure of any two quantities: First, When it is a simple quantity. To the greatest common measure of the numerical coefficients annex the common letters. Second, When it is a compound quantity. (1.) Divide the one given quantity by the other, using as divisor the one which is of lower dimensions, if the dimensions be different. (2.) If there be a remainder, divide the divisor by it. (3.) If there still be a remainder, divide the last divisor by it; and continue the process till nothing remain. The last divisor is the Greatest Common Measure. This rule depends on the two following principles : 1. If one quantity measure another, it will measure any multiple of it. 2. If one quantity measure two others, it will measure their sum or difference. In order to simplify the operation, either of the original quantities, or any of the remainders, must be divided by any |