From (a + b + c), take a, b, c, separately and successively. Subtract − 3 √a + ≈ + 4 (x2 — y2) — 1, from ‚/x2 + y2—2 (a+x)2 + 3; - x − and 17axy + 11abc — x'√x + y, from 2x (x + 3)* c, from ax2+mxy + nx + b; and a), from · y)2 + bxy + c (a take (a - b) (x + y) + (c (x + y) + m. pxy + qrx + x)2, take (n — y) r (22 + a). ·bxy + (a + c) (a + x)2; and d) (x − y) —n, from (a + b) (x + y) − (c + d) From (a—b) xy—hæ2 subtract (2p—3q) (x+y)*; and from — (p + q) √x + y› This consists of several cases, according as the factors are simple or compound quantities. CASE I. When both the factors are simple quantities. 1. MULTIPLY the co-efficients of the two terms together; then, to the product annex all the letters in those terms, which will give the whole product required. 2. When the same letter is repeated in the product, the result may be simplified by writing the sum of the indices instead of the separate factors, agreeably to def. p. 105. Thus, for a3 × a2, write a3. 3. The same rule holds good if there be fractional indices, or their equivalent radicals, in the product: as for a1a, write a +, or a; and for a xa, write a1 or a. This is true whatever a may represent, as for instance, if a = (−1)* × (−1) * or √1x-1=-1. 1)* 1, then Note*. Like signs, in the factors, produce +, and unlike signs —, in the products. *That this rule for the signs is true, may be thus shown. 1. When a is to be multiplied by +c; the meaning is, that a is to be taken as many times as there are units in c; and since the sum of any number of positive terms is positive, it follows that a x + c makes + ac. 2. When two quantities are to be multiplied together, the result will be exactly the same, in whatever order they are placed; for a times c is the same as c times a, and therefore, when Though only two factors have been proposed, there may be any number. The process is however still the same, repeating the operation with every successive term upon the result of all the preceding. When, however, the factors are all equal, the literal parts may be more readily assigned; viz. multiply the index of each letter in the common factor by the index of the number of factors. The products of these are the indices of the literal parts. Thus, am b” c3 × am b” c2 × am b2 c2 = a3m b3n c3r, or (am b2 co)3. When there are numeral coefficients, the powers of these must be found as at p. 65. Thus, suppose we sought the products of The signs being regulated by the number of factors when those factors are —. This is a case of INVOLUTION. a as many is to be multiplied by + c, or + c by a this is the same thing as taking times as there are units in + c; and as the sum of any number of negative terms is negative, it follows that a x + c, or + ax c make or produce · -ac. 3. When a is to be multiplied by - -c: here -a is to be subtracted as often as there are units in c but subtracting negatives is the same thing as adding affirmatives, by the demonstration of the rule for subtraction; consequently, the product is c times a, or + ac. Otherwise. Since a a= 0, therefore (aa) × - c is also = 0, because 0 multiplied by any quantity, is still but 0; and since the first term of the product, or a × — c is —— ac, by the second case; therefore the last term of the product, or — a x c, must be ac, to make the sum = = 0, or - ac + ac = 0; that is, -ax - cac. Other demonstrations upon the principles of proportion, or by means of geometrical diagrams, have also been given; but the above is more natural, simple, and satisfactory. 5. To find the 6th power of ± 2a2 √ ± 1. 6. To find the 7th power of (+1)3 × (± a2) (— a−3). CASE II. When one of the factors is a compound quantity. MULTIPLY every term of the multiplicand, or compound quantity, separately, by the multiplier, as in the former case; placing the products one after another, with the proper signs; and the result will be the whole product required. When both the factors are compound quantities. MULTIPLY every term of the multiplier by every term of the multiplicand separately; setting down the products one after or under another, with their proper signs; and add the several partial products together for the whole product required. or 3a Note I. When the factors are all equal, and composed of two terms, as a + x, 5x, the operation is more readily performed by the BINOMIAL THEOREM, the rule of which may be referred to at once. For any purposes likely to occur in the earlier stages of Algebra, the result may be obtained by actual multiplication, as above. When there are more than two terms, there is a general theorem for finding the coefficients, called the MULTINOMIAL THEOREM: but as occasion for its use occurs comparatively seldom, it will not be given in this work. Reference may therefore be made to Young's Algebra, p. 262. When the factors are all equal, as we have here supposed, the operation is called INVOLUTION. 3xy to the fourth power. Ex. Raise a x to the third power, and 2a Note II. The following is a specimen of the method of disposing of the literal coefficients in vertical columns. It has not only the advantage of keeping an operation of considerable extent within the limits of the breadth of the page, but it dispenses with the collecting those coefficients together, after the multiplications are developed, on account of its readily disposing them in their places as we proceed. Multiply together the binomials a b, x -C. which in the horizontal disposition of the terms of the coefficients would stand thus: 23 − (a + b + c) x2 + (ab + bc + ca) x - abc. Had the number of factors been greater, the advantage would have been still more apparent. In the case just considered, where the given quantities are numerically given, this disposition of them is the most favourable to their actual addition into one numerical sum. It is not necessary to write down the powers of the quantity according to which the work is arranged, till we have performed the whole of the arithmetical determination of the coefficients: since the same powers of that letter, if generated by the multiplication of factors in which none of the terms are wanting, will always, in the above arrangement, fall in the same vertical column. Also, since the indices of the powers of that letter in going from term to term, either decrease by units or increase by units, according as we begin at the highest or lowest powers, we may write them in juxtaposition with the coefficients found as above indicated, without the slightest degree of attention beyond the most ordinary kind. In finite expressions, it is most usual, though not essential, to begin with the highest, and in series, it is, from their interminable character, necessary to begin with the smallest index, whether positive or negative. The following example will render the practice obvious: and the student is earnestly recommended to adopt it, not only on account of economy of time and room; but to familiarize his hand, his eye, and his thoughts, to the processes by which the roots of equations are (in the most improved state of that difficult problem) found or approximated to. Here the coefficients of a2 in the first, and of a in the second factor, are each equal to zero. Hence the work will stand thus: And the highest power of a being a3 × a2 = a5, we may write a3, a1, a3, a2, a1, in juxtaposition with the above coefficients, which gives for the product The same result would have been obtained, but in an inverted order, by writing (15 · 4a + 2a3) (4 + 3a2). When the first term of the multiplier is also unity, the work may be shortened still more, by allowing the line of coefficients which constitute the multiplicand, as the first partial product of the work. Thus, were ≈3 6x2+10x 9 given to be multiplied by x2 3x+2; the general method would require it to be executed as follows. 18 And x3. x2 = ∞ is the highest power or first term. Whence, attaching the literal parts, we get as the product But as the line (b) is the same with the line (a) we may put it thus: * Here, instead of a horizontal column of ciphers, the first term of the multiplication by 4 is made to commence under the 4, as in common arithmetical multiplication. |