And since the same course of reasoning may be applied to any two monomials, we have, for the multiplication of mono mials, the following RULE. 1. Multiply the co-efficients together for a new co-efficient. 11. Write after this co-efficient all the letters which enter into the multiplicand and multiplier, giving to each an exponent equal to the sum of its exponents in both factors. 43. We will now proceed to the multiplication of polynomials. In order to explain the most general case, we will suppose the multiplicand and multiplier each to contain additive and subtractive terms. Let a represent the sum of all the additive terms of the multiplicand, and — b the sum of the subtractive terms; c the sum of the additive terms of the multiplier, and d the sum of the subtractive terms. The multiplicand will then be represented by ab and the multiplier, by cd. We will now show how the multiplication expressed by ·(a - b) × (c — d) can be effected. e times; for it is only the difference between a and b, that is first to be multiplied by c. Hence, ac be is the product of a-b by c. b taken c d times: hence, the But the true product is a last product is too great by a b taken d times; that is, by ad bd, which must, therefore, be subtracted. Saptracting this from the first product (Art. 37), we have If we suppose a and c each equal to 0, the product will re duce to bd. 44. By considering the product of ab by cd, we may deduce the following rule for signs, in multiplication. When two terms of the multiplicand and multiplier are affected with the same sign, their product will be affected with the sign +, and when they are affected with contrary signs, their product will be affected with the sign -. We say, , in algebraic language, that + multiplied by + or multiplied by -, gives +; - multiplied by +, or + mul tiplied by gives-. But since mere signs cannot be multiplied together, this last enunciation does not, in itself, express a distinct idea, and should only be considered as an abbreviation of the preceding. This is not the only case in which algebraists, for the sake of brevity, employ expressions in a technical sense in order to se cure the advantage of fixing the rules in the memory. 45. We have, then, for the multiplication of polynomials, the following RULE. Multiply all the terms of the multiplicand by each term of the multiplier in succession, affecting the product of any two terms with the sign plus, when their signs are alike, and with the sign minus, when their signs are unlike. Then reduce the polynomial result to its simplest form. 6. Multiply 3ab2 + 6a2c2 by 3ab2 + 3a2c2. 7. Multiply 4x2 - 2y by 2y. Ans. 9a2b4+ 27a3b2c2 + 18a*c*. 8. Multiply 2x + 4y by 2x-4y. Ans. 8x2y4y2. Ans. 4x2 - 16y2. 9. Multiply 23 + x2y + xy2 + y3 by x-y. 10. Multiply 2 + xy + y2 by x2 — xy + y2. Ans. Ans. x + x2y2 + ya. In order to bring together the similar terms, in the product o two polynomials, we arrange the terms of each polynomial wito reference to a particular letter; that is, we arrange them so tha the exponents of that letter shall go on diminishing from left to right. After having arranged the polynomials, with reference to the letter a, multiply each term of the first, by the term 2a2 of the second; this gives the polynomial 8a5 10ab16a3b2+4a2b3, in which the signs of the terms are the same as in the multiplicand. Passing then to the term 3ab of the multiplier, mul tiply each term of the multiplicand by it, and as it is affected with the sign, affect each product with a sign contrary to that of the corresponding term in the multiplicand; this gives 12ab+15a3b2 + 24a2b3 - 6ab1. Multiplying the multiplicand by The product is then reduced, and we finally obtain, for the most simple expression of the product, 8a5 22a+b 17a36248a2b3 + 26ab1 865. - 12. Multiply 2a2 - 3ax + 4x2 by 5a2 6ax 2x2. Ans. 10a 27a3x + 34a2x2. - 13. Multiply 3x2 - 2yx + 5 by x2 + 2xy - 3. 18αx3 8x+. Ans. 3x+4x3y — 4x2 - 4x2y2 + 16xy — 15. 14. Multiply 3x3 + 2x2y2+3y2 by 2x3-3x2y2+5y3. - 6x6 — 5x3y2 — 6x1y1 + 6x3y2 + 15x3y* − 9x2y+ + 10x2y5 + 15y3. -c by 2ax +ab + e. 4a2bx 6a2b26acx — Tabc - c2. -- 15a37a2bd29a2cf — 20b2d2+44bcdf8c2f 18 Multiply 4a3b? - 5a2b2c + Sa2bc2 3a2c3abc3 46. REMARKS ON THE MULTIPLICATION OF POLYNOMIALS. 1st. If both multiplicand and multiplier are homogeneous, the product will be homogeneous, and the degree of any term of the product will be indicated by the sum of the numbers which indicate the degrees of its two factors. Thus, in example 18th, each term of the multiplicand is of the 5th degree, and each term of the multiplier of the 3d degree hence, each term of the product is of the 8th degree. This remark serves to discover any errors in the addition of the exponents. 2d. If no two terms of the product are similar, there will be no reduction amongst them; and the number of terms in the product will then be equal to the number of terms in the multiplicand, multi plied by the number of terms in the multiplier. This is evident, since each term of the multiplier will produce as many terms as there are terms in the multiplicand. Thus, in example 16th, there are three terms in the multiplicand and two in the multiplier: hence, the number of terms in the product is equal to 3 × 2 = 6. 3d. Among the terms of the product there are always two which cannot be reduced with any others. For, let us consider the product with reference to any letter common to the multiplicand and multiplier: Then the irreduci ble terms are, 1st. The term produced by the multiplication of the two terms of the multiplicand and multiplier which contain the highest power of this letter; and 2d. The term produced by the multiplication of the two terms which contain the lowest power of this letter. For, these two partial products will contain this letter, to a higher and to a lower power than either of the other partial pro ducts, and consequently, they cannot be similar to any of them, This remark, the truth of which is deduced from the law of the exponents, will be very useful in division. |