14. What is the continued product of 5a, 4m2x, and 9a2m3x? Ans. 180a3m5x2 15. What is the continued product of 7a2b, ab3, and 4ac3? 16. What is the continued product of 3amx, 5ab2, and 7abx? 17. What is the continued product of a, ab, abc, abcd, and abcdx? 18. What is the continued product of a3, a3b2, a3bc, and a365c1x? CASE II. When one or both of the factors are polynomials. 60. Represent the sum of the positive terms of any polyno mial by a, and the sum of the negative terms by b. Then a-b will represent any polynomial whatever. In like manner, c-d will represent any other polynomial whatever. required to find the product of a-b by c—d. It is This implies In the first place, let us multiply a−b by c. that the difference of the units in a and b is to be repeated c times. If a be repeated as many times as there are units in c, the result will be +ac. Also, if -b be repeated as many times as there are units in c, the result will be taken twice is 2b, taken three times is -36, etc.; repeated c times, the result will be -cb or -bc. operation may be exhibited thus: bc, for b and if it be The entire a-b с ac-bc. Next let us multiply a-b by c-d. When we multiply a-b by c, we obtain ac-bc. But a-b was only to be taken c-d times; therefore, in this first operation, we have repeated it d times more than was required. Hence, to have the true product, we must subtract d times a-b from ac-bc. But d times a-b is equal to ad—bd, which, subtracted from ac—bc, gives ac-bc-ad+bd. If the pupil does not perceive the force of this reasoning, it will be best to repeat the argument with numbers, thus: Let it be proposed to multiply 8-5 by 6-2; that is, the quantity 8-5 is to be repeated as many times as there are units in 6-2. If we multiply 8-5 by 6, we obtain 48-30; that is, we have repeated 8-5 six times. But it was only required to repeat the multiplicand four times, or 6-2. We must therefore diminish this product by twice 8–5, which is 16—10; and this subtraction is performed by changing the signs of the subtrahend. Hence we have 48-30-16+10, which is equal to 12. This result is obviously correct; for 8-5 is equal to 3, and 6—2 is equal to 4; that is, it was required to multiply 3 by 4, the result of which is 12, as found above. We have thus obtained the following results: +ax(+b)=+ab, +ax(—b)=—ab, —a×(+b)=—ab, —a×(—b)=+ab, from which we perceive that when the two factors have like signs, the product is positive; and when the two factors have unlike signs, the product is negative. 61. Hence, for the multiplication of polynomials, we have the following general RULE. Multiply each term of the multiplicand by each term of the mul tiplier, and add together all the partial products, observing that like signs require + in the product, and unlike signs 2. a2+2ab+b2 a3+2a2b+ ab2 a2b+2ab2+b3 a3+3a2b+3ab2+b3 It is immaterial in what order the terms of a polynomial are arranged, or in what order the letters of a term are arranged. It is, however, generally most convenient to arrange the letters of a term alphabetically, and to arrange the terms of a polynomial in the order of the powers of some common letter. 3. Multiply a2—ab+b2 by a+b. 4. Multiply a2-2ab+l2 by a−b. 5. Multiply 3a2-2a+5 by a-4. Ans. a3+b3. 6. Multiply a2-ab+b2 by a2+ab+b2. Ans. a*+a2b2+ba. 7. Multiply 2a2-3ab+4 by a2+2ab—3. 8. Multiply a3+a2b+ab2+b3 by a-b. 9. Multiply a+mb by a+nb. 10. Multiply 3a+2bx-3x2 by 3a-2bx+3x3. 11. Multiply together x-5, x+2, and x+3. 12. Multiply together x-3, x-4, x+5, and x-6. 13. Multiply together a2+ab+b2, a2—ab+b2, and a2-b2. 14. Multiply together a+x, b+x, and c+x. 15. Multiply a+a3b+a2b2+ab3+ba by a−b. 16. Multiply a3-3a2+7a-12 by a2+3a+2. 17. Multiply x+2x3+3x2+2x+1 by x2-2x+1. 18. Multiply 14a3x-6a2bx+x2 by 14a3x+6a2bx-x2. 19. Multiply 3—x2y+xy2 by x2—xy2—y3. 20. Multiply 3x2+8xy-5 by 4x2—7xy+9. 62. Degree of a Product.-Since, in the multiplication of two monomials, every factor of both quantities appears in the prod uct, it is obvious that the degree of the product will be equal to the sum of the degrees of the multiplier and multiplicand. Hence, also, if two polynomials are homogeneous, their product will be homogeneous. Thus, in the first example of the preceding article, each term of the multiplicand is of the first degree, and also each term of the multiplier; hence each term of the product is of the second degree. For a similar reason, in the second example, each term of the product is of the third degree; and in the sixth example, each term of the product is of the fourth degree. This principle will assist us in guarding against errors in the multiplication of polynomials, so far as concerns the exponents. 63. Number of Terms in a Product.-When the product arising from the multiplication of two polynomials does not admit of any reduction of similar terms, the whole number of terms in the product is equal to the product of the numbers of the terms in the two polynomials. Thus, if we have five terms in the multiplicand, and four terms in the multiplier, the whole number of terms in the product will be 5×4, or 20. In general, if there be m terms in the multiplicand, and n terms in the multiplier, the whole number of terms in the product will be mxn. 64. Least Number of Terms in a Product.-If the product of two polynomials contains similar terms, the number of terms in the product, when reduced, may be much less than mn; but it is important to observe that among the different terms of the product there are always two which can not be combined with any others. These are, 1st. The term arising from the multiplication of the two terms affected with the highest exponent of the same letter. 2d. The term arising from the multiplication of the two terms affected with the lowest exponent of the same letter. For it is evident, from the rule of exponents, that these two partial products must involve the letter in question, the one with a higher, and the other with a lower exponent than any of the other partial products, and therefore can not be similar to any of them. Hence the product of two polynomials can never contain less than two terms. 65. For many purposes it is sufficient merely to indicate the multiplication of two polynomials, without actually performing the multiplication. This is effected by inclosing the polynomials in parentheses, and writing them in succession, with or without the sign x. When the indicated multiplication has been actually performed, the expression is said to be expanded. EXAMPLES. 1. Expand (a+b) (c+d). 2. Expand 9a-7(b—c). 3. Expand and reduce Ans. ac+be+ad+bd. 14(12-a-b-c)+13(4+a−c)—15(7—a—c). 4. Expand and reduce 28(a-b+c)+24 (a+b-c)-13 (b-a-c). 5. Expand and reduce 24a-6b-9(a+b)+25a—19(b−c)—17(a+b−c). 6. Expand and reduce 53(a−b+c)−27(a+b−c)−26 (a−b—c). 66. The three following theorems have very important applications. The square of the sum of two numbers is equal to the square of the first, plus twice the product of the first by the second, plus the square of the second. by Thus, if we multiply we obtain the product a+b Hence, if we wish to obtain the square of a binomial, we can, according to this theorem, write out at once the terms |