CHAPTER IV. MULTIPLICATION. expressions. 53. Product of monomial The multiplication of monomial expressions was considered in Chapter II., and the results arrived at were: (i) The factors of a product may be taken in any order. (ii) The sign of the product of two quantities is + when both factors are positive or are both negative; and the sign of the product is when one factor is positive and the other negative. (iii) The index of the product of any two powers of the same quantity is the sum of the indices of the factors. From (i), (ii) and (iii) we can find the continued product of any number of monomial expressions. Thus (-2a2bc3) × ( − 3a3b2c) = +2a2bc3 × 3a3b2c, from (ii), =2×3× a2. a3. b. b2. c3. c, =6a5b3c4, from (i), from (iii). Again, (-3a2b) (-5ab3) (-7a4b2)= {+3a2b. 5ab3} (-7a4b2) 54. Product of a multinomial expression and a monomial. It was proved in Art. 42 that the product of the sum of any two algebraical quantities by a third is equal to the sum of the products obtained by multiplying the two quantities separately by the third. Thus (x + y) z = xz + yz........ .(i). Since (i) is true for all values of x, y and 2, it will be true when we put (a + b) in place of x; hence {(a+b)+ y} z = (a + b) z + yz = az + bz+yz. :. (a+b+ y) z = az + bz + yz. And similarly (a+b+c+d+ ...) z = az + bz + cz + dz + however many terms there may be in the expression a+b+c+d+... Thus the product of any multinomial expression by a monomial is the sum of the products obtained by multiplying the separate terms of the multinomial expression by the monomial. 55. Product of two multinomial expressions. We now consider the most general case of multiplication, namely the multiplication of any two multinomial expressions. We have to find (a + b + c + ...) × (x + y + z + ...) ; and, from Art. 38, this includes all possible cases. (a+b+c+...) M = aM+bM+ cM + ... = (x + y + 2 + ...) a + (x + y + z + ...) b + (x + y + z + ...) C + ... = ax+ay + az + ... + bx+by+ bz + ... + cx + cy + cz + ... Hence (a + b + c + ...) (x + y + z + ...) = ax+ay + az + ... + bx + by + bz + ... + cx + cy+cz + ... Thus the product of any two algebraical expressions is equal to the sum of the products obtained by multiplying every term of the one by every term of the other. also For example (a + b) (c + d) = ac + ad + bc + bd ; (3a +5b) (2a+3b) = (3a) (2a) + (3a) (3b) + (5b) (2a) +(5b) (3b)=6a2+9ab+ 10ab + 15b 6a+ 19ab + 1562. = Again, to find (a - b) (c - d), we first write this in the form {a+(-b)} {c + (− d)}, and we then have for the product ac+a (d)+(-b) c + (-b) (− d) = ac-ad-bc + bd. In the rule given above for the multiplication of two algebraical expressions it must be borne in mind that the terms include the prefixed signs. 56. The following are important examples : : I. (a+b)2 = (a + b) (a + b) = aa + ab + ba +bb; Thus, the square of the sum of any two quantities is equal to the sum of their squares plus twice their product. II. (a - b)2 = (a − b) (a − b) = aa + a (− b) + (− b) a +(-b) (-b) = a2 — ab — ab + b2; Thus, the square of the difference of any two quantities is equal to the sum of their squares minus twice their product. III. (a+b) (a - b) = aa + a (− b) + ba + b (− b) Thus, the product of the sum and difference of any two quantities is equal to the difference of their squares. S. A. 3 57. It is usual to exhibit the process of multiplication in the following convenient form: The multiplier is placed under the multiplicand and a line is drawn. The successive terms of the multiplicand, namely a2, +2ab, and - b2, are multiplied by a2, the first term on the left of the multiplier, and the products a*, +2ab and - a2b2 which are thus obtained are put in a horizontal row. The terms of the multiplicand are then multiplied by - 2ab, the second term of the multiplier, and the products thus obtained are put in another horizontal row, the terms being so placed that 'like' terms are under one another. And similarly for all the other terms of the multiplier. The final result is then obtained by adding the rows of partial products; and this final sum can be readily written down, since the different sets of 'like' terms are in vertical columns. The following are examples of multiplication arranged as above described: 58. If in an expression consisting of several terms which contain different powers of the same letter, the term 3x2- xy+2y2 9x4-3x3y+6x2y2 9.x4 x2y2 + 4xy3 — 4y1 which contains the highest power of that letter be put first on the left, the term which contains the next highest power be put next, and so on; the terms, if any, which do not contain the letter being put last; then the whole expression is said to be arranged according to descending powers of that letter. Thus the expression a3 + a2b+ ab2 + b3 is arranged according to descending powers of a. In like manner we say that the expression is arranged according to ascending powers of b. 59. Although it is not necessary to arrange the terms either of the multiplicand or of the multiplier in any particular order, it will be found convenient to arrange both expressions either according to descending or according to ascending powers of the same letter: some trouble in the arrangement of the different sets of 'like' terms in vertical columns will thus be avoided. 60. Definitions. A term which is the product of n letters is said to be of n dimensions, or of the nth degree. Thus 3abc is of three dimensions, or of the third degree; and 5a b'c, that is 5aaabbc, is of six dimensions, or of the sixth degree. Thus the degree of a term is found by taking the sum of the indices of its factors. The degree of an expression is the degree of that term of it which is of highest dimensions. In estimating the degree of a term, or of an expression, we sometimes take into account only a particular letter, or particular letters: thus ax + bx + c is of the second degree in x, and ax3y + bxy + cx2 is of the third degree in x and y. An expression which does not contain x is said to be of no degree in x, or to be independent of x. When all the terms of an expression are of the same dimensions, the expression is said to be homogeneous. Thus a3ab - 56 is a homogeneous expression, every term being of the third degree; also ax2 + bxy + cy2 is a homogeneous expression of the second degree in x and y. |