by ax. by 3xz. by a2 x3. 4. Divide 17 a m2 x3 5. Divide 21 x3 y4 z 6. Divide a2 x3 7. Divide 35 x7 y1 8. Divide 350 m2 y6 9 Divide 64 a3 m5 x 10. Divide 33 x9 y7 z6 11. Divide 150 a m3 x 4 12. Divide 17 ax 13. Divide 18 a2 b3 c2 x 14. Divide 28 m x t p2 15. Divide 333 a7 x y1 z 16. Divide 111 p5 x2 y5 17. Divide 99 a1 18. Divide 17 a m1 x 4 19. Divide 3 a5 b4 c3 x7 20. Divide 75 m10 y9 by 7 x y3. by 6ab2 c. by 4m xp. by 25 m2 y6. by 5 x3 y2 z. by abc. by 4 m3 x y2. SECTION XIII. ART. 49. DIVISION OF POLYNOMIALS. If a+m-c be multiplied by d, the product will be ad+dm-cd; consequently, if a d+ dm-cd be divided by d, the quotient will be a+m—c. This quotient is obtained by dividing each term of the dividend, ad+dm-cd, by the divisor d. In order, therefore, that a polynomial may be divisible by a monomial, each term of the dividend must be divisible by the divisor. The rule for the signs of the partial quotients may be determined as follows, if we recollect that, in all cases, the product of the divisor and quotient must give the dividend. If+am be divided by +a, the quotient must be +m, because the product of +a and +m is + am. If+am be divided by ―a, the quotient must be m, because the product of - a and—m is +am. m, If — am be divided by +a, the quotient must be because the product of +a and —m is -am. Finally, if -am be divided by - a, the quotient must be +m, because the product of -a and +m is am. We perceive, therefore, that when two terms, whose quotient is sought, have the same sign, whether both + or both —, the quotient must have the sign +; but when the signs of the two terms are different, the quotient must have the sign ART. 50. From what precedes, we derive the following RULE FOR THE DIVISION OF A POLYNOMIAL BY A MONOMIAL. 1. Divide each term of the dividend by the divisor, according to the rule for dividing one monomial by another; the partial quotients, taken together, will form the entire quotient. 2. With regard to the signs, observe that when the term of the dividend and the divisor have the same sign, the corresponding term of the quotient has the sign +; but when the term of the dividend and the divisor have dif ferent signs, the corresponding term of the quotient has the sign 1 Divide am+m2 x by m. 2. Divide 3x2y+6x by 3x. 3. Divide 15 m y2-30 y3 by 5 yo. Ans a+mx. 4. Divide 45 m2 — 15 m3 y+5m by 5m. 9. Divide 40 x2 +50 xy-30 x y2 by -5x. 10. Divide 10 abc-16 ab2-20 a3 b by -2 ab. 11. Divide 16 m x2+ 32 m2 x-24 m3 x2 by 8mx. 12. Divide 45 a2x3+3 a3 x2-60 at x2 by -3 a2x2. ART. 51. When both dividend and divisor are polynomials, the process of dividing will be easily understood, if we observe the formation of a product in multiplication. Let us multiply 3 a2+2 a b by 4a+3b. Operation. 3a2+2 a b 12 a38 a2 b +9a2 b+6 a b2 12 a3 17 a2 b+6 a b2. If this product be divided by the multiplicand, the quotient will be the multiplier; or if it be divided by the multiplier, the quotient will be the multiplicand. Since, in multiplying, the entire multiplicand is multiplied by each term of the multiplier, the product, if no reduction took place, would contain a number of terms equal to the product obtained by multiplying the number of terms in the multiplicand by the number of terms in the multiplier. Thus, in the preceding example, there are 2 terms in the multiplicand, and 2 in the multiplier, and, if 8 a2b and 9 a2b had not been united, there would have been 2.2 or 4 terms in the product. In like man`ner, if one factor had 5 terms and the other 3, the product, without reduction, would contain 5.3 or 15 terms. But generally, by reduction, some terms of the product are united, and others are cancelled and disappear. There are, however, two terms of the product which can neither be united with others, nor disappear. These are, 1st, the product of the term containing the highest power of any letter in the multiplicand by the term containing the highest power of the same letter in the multiplier; 2d, the product of the two terms containing the lowest powers of the same letter. Since the dividend is to be considered as the product of the divisor and quotient, it is plain that if the term containing the highest power of any letter in the dividend be divided by the term containing the highest power of the same letter in the divisor, the result will be the term containing the highest power of that letter in the quotient. Let us now take the product of the preceding multipli cation for a dividend, and the multiplicand for a divisor, and see how we can obtain the multiplier, considered as a quotient. -According to what has been said above, if we divide 12 a3 by 3 a2, we shall obtain the term of the quotient with the highest power of a. This division gives 4 a for the first term of the quotient. Then, since the entire dividend is produced by multiplying the whole divisor by each term of the quotient, if we multiply the whole divisor by 4 a, this first term of the quotient, the product will be a part of the dividend. This product is 12 a3+8a2b, which we subtract from the dividend, and find for a remainder 9 ab +6 a b2. This remainder is to be considered as a new dividend, and as produced by multiplying the divisor by the remaining part of the quotient; and if the term containing the highest power of a in this new dividend be divided by the term containing the highest power of a in the divisor, the result must be a new term of the quotient. The division of 9 a2b by 3 a2 gives +3b, which we write as the second term of the quotient. We now multiply the whole divisor by 3b, and the product must be the whole or a part of the new dividend. This product is 9a2b+6ab2, which, subtracted from the new dividend, leaves no remainder. The entire quotient, therefore, is 4a+3b. Since we always divide the term containing the highest power of some letter in the dividend by the term containing the highest power of the same letter in the divisor, it is convenient to write the quantities so that the former of these terms shall stand first in the dividend, and the latter first in the divisor. This object will be attained by arranging both dividend and divisor according to the powers of the same letter. Remark. A polynomial is said to be arranged according to the powers of a particular letter, when, in the successive terms, the powers of that letter increase or |