polynomials is to be added to the product of the first and second. 7. (3 m2 + 3 m n) (3x+2y) + (5 m2 + 4 a x) (2 x -y). 8. (17 x2 — 2y) (a2 + 2 b) + (25 x2 + 3 y) (3 a2 5b). - 9. (6xy+3az) (2 x2 — 5 a z) — (2 x y — m x) (2x2 -4m - 4 mx). In the 9th example, the product of the last two quantities is to be subtracted from that of the first two. 10. (3abcd) (m2 x+ny) — (3 m2 x-4 ny) 2ab. 11. (a5+5 ax+2y2) b2 c— (a1 +4 a3 x) (a b2 c — m y2). 12. (5 a2 — 3 a b + 4 b2) (6 a — 5 b) — 3 a (a2+ab). ART. 44. The following examples deserve particular attention, on account of the use which will hereafter be made of the results. The sum of the two quantities, a and b, is a+b, and their difference is a-b. Let us multiply this sum by this difference. This product is the difference between the second power of a, and the second power of b. Hence, The sum of two quantities multiplied by their difference gives the difference of the second powers of those quantities According to this principle, (8+3) (8—3) — 64 — 9 =55, and (3a+x) (3 a-x)=9 a2x2. = Let the learner give the results of the following examples, without actually multiplying. 1. (x+y) (xy). 2. (m+n) (m—n). 3. (3a-4x) (3 a +4x). 4. (4a2+3y) (4 a2 — 3 y). 6. (6 a2b—3 m) (6 a2b+3m). ART. 45. When a polynomial is multiplied by itself, the product is called the second power, and when multiplied twice by itself, the product is called the third power of that polynomial. Find the second power of a+b. a+b a+b Operation. a2+ab +ab+b2 a2+2ab+b2=2d power of a+b. Hence the second power of the sum of two quantities contains the second power of the first quantity, plus twice the product of the first by the second, plus the second power of the second. Find the second power of a-b. This result differs from the second power of a+b only in the sign of 2 ab, which is in this case negative. By means of the two principles in this and the preceding article, write the second powers of the following quantities. ART. 46. If the second power of a+b is multiplied by a+b, the product will be the third power of a+b,* which is a3+3a2b+3 a b2+b3. Hence the third power of the sum of two quantities contains the third power of the first quantity, plus three times the second power of the first into the second, plus three times the first into the second power of the second, plus the third power of the second. In like manner, the third power of a to be a3-3a2b+3 a b2 — b3, b will be found which differs from that of a+b, in having its second and fourth terms negative. Write the third powers of the following quantities. 1. x+y. 2. x- -y. 3. m+n. 4. m-n. 5. 2a+m. 6. 4 x -y. 7. 3m+2x. 8. 2m2-3y2 SECTION XII. DIVISION OF MONOMIALS. ART. 47. Division being the reverse of multiplication, the object is to find, for the quotient, a quantity, which, when multiplied by the divisor, shall produce the Jividend in other words, having a product and one of its factors, the object is to find the other factor. Since, then, the product of the divisor and quotient must reproduce the dividend, the coefficient of the quotient must be such, that, when multiplied by the coefficient of the divisor, it shall produce that of the dividend; and the exponent of any letter in the quotient must be such, that, when added to the exponent of the same letter in the divisor, it shall give the exponent of that letter in the dividend. Also, it is manifest that the quotient must contain those letters of the dividend which are not found in the divisor. The answer is a; Divide ma by m, or find m of ma. because, if m be multiplied by a, the product is ma. Ans. 3 a. We readily see the correctness of the preceding answers, because, in each case, the product of the divisor and quotient gives the dividend. Divide a5 by a2. The quotient is a3, because a2. a3 =a5. Divide 7 a2b5c by 7ab3. The quotient is ab2c, because 7ab3. a b2 c=7a2 b5 c. We found, in the multiplication of monomials, that, when the same letter occurred in both multiplicand and multiplier, the exponents were added, to obtain the exponent of that letter in the product. On the other hand, in dividing, as we see in the last two examples, the exponent of any letter in the divisor is subtracted from the exponent of the same letter in the dividend, in order to obtain the exponent of that letter in the quotient. Thus, a5, divided by a2, gives a5—2— a3. ART. 48. From what precedes, we deduce the following RULE FOR DIVIDING ONE MONOMIAL BY ANOTHER. 1. Divide the coefficient of the dividend by the coefficient of the divisor. 2. Strike out from the dividend the letters common to it and the divisor, when they have the same exponents in both; but if the exponents of any letter are different, subtract its exponent in the divisor from its exponent in the dividend, and write the letter in the quotient with the remainder for an exponent. 3. Write also in the quotient, with their respective exponents, the letters of the dividend not found in the divisor. Remark. If, however, the divisor and dividend are, in any case, alike, the quotient is 1. 1. Divide 6 a2b3 by 2 ab. 2. Divide 25 a1 by 5 a2. Ans. 3 a b2. |