-8x2y-8a+7b6ax-2x2y +5xy2 -x2+8x+2x-4y EXAMPLES FOR PRACTICE. 1. Find the difference of (a + b) and (a - b). Ans. b. 2. From 3x - 2a - b + 7, 7, take 8-36 + a + 4x. 3. From 3a + b + c - 2d, take b Ans. 26- х 8c+2d-8. Ans. За -1. 3a +9c-4d + 8. 1 - а. 4. From 13x2-2ax + 962, take 5x2 Ans. 5. From 20aах - 5√x + 3a, take 4ax + 5x2 Ans. 16ax 10√x + 4a. 6. From 5ab +262-c+bc-b, take b2 - 2ab + bc. Ans. 7ab+b2 — c —b. 7. From ax3-bx2 + cx-d, take bx2 + ex - 2d. Ans. ax3 - 2bx + (c - e) x + d.. 8. From -6a-46-12c + 13x, take 4x - 9а + 46-5c. Ans. 3a + 9x - 8b -7c. 9. From 6x2y-3√(xy) - 6ay, take 3x2y + 3 (xy) - 4ay. Ans. 3x2y- 6√(xy) - 2ay. 10. From the sum of 4ax - 150+4x2, 5x2 + 3ax +10x2, and 90 - 2ax - 12√(x); take the sum of 2ax - 80 + 7x2, - 8ax-70, and 30-4√(x) - 2x2 + 4a2x2. 1 1 7x2 I Ans. 11ax +60 - x2 - 4a2x2. MULTIPLICATION. MULTIPLICATION, or the finding of the product of two or more quantities, is performed in the same manner as in arithmetic; except that it is usual, in this case, to begin the operation at the lefthand, and to proceed towards the right, or contrary to the way of multiplying numbers. The rule is commonly divided into three cases; in each of which it is necessary to observe, that like signs, in multiplying, produce +, and unlike signs, -. It is likewise to be remarked, that powers, or roots of the same quantity, are multiplied together by adding their indices: thus, I a × a2, or a × a2 = a2; a2 × a2 = a2; a2 xa am Xan = am + n. 1 3 a; and The multiplication of compound quantities, is also, sometimes, barely denoted by writing them down, with their proper signs, under a vinculum, without performing the whole operation, as 3ab (a - b), or 2a√(a2 + b2). Which method is often preferable to that of executing the entire process, particularly when the product of two or more factors is to be divided by some other quantity, because, in this case, any quantity that is common to both the divisor and dividend, may be more readily suppressed; as will be evident from various instances in the following part of the work.* CASE I. When the factors are both simple quantities. RULE.-Multiply the coefficients of the two terms together, and to the product annex all the letters, or their powers, belonging to each, after the manner of a word; and the result, with the proper sign prefixed, will be the product required.t * The above rule for the signs may be proved thus: If B, b, be any two quantities, of which B is the greater, and B-b is to be multiplied by a, it is plain that the product, in this case, must be less than ав, because B-b is less than B; and, consequently, when each of the terms of the former are multiplied by a, as above, the result will be (в-б) х а = ав - ав. For if it were ав + ab, the product would be greater than as, which is absurd. Also, if a be greater than b, and a greater than a, and it is required to multiply B-8 by a - a, the result will be (B-b) × (-a) = AB- aB-ba+ab. For the product of B-6 by a is a (B-6), or AB - Ab, and that of B-6 by-a, which is to be taken from the former, is - а (в -6) as has been already shown; whence B - b being less than B, it is evident that the part, which is to be taken away must be less than ав; and consequently since the first part of this product is ав, the second part must be + ab; for if it were - ab, a greater part than as would be to be taken from A (B-6), which is absurd. + When any number of quantities are to be multiplied together, it is the same thing in whatever order they are placed: thus, if ab is to be When one of the factors is a compound quantity. RULE.-Multiply every term of the compound factor, considered as a multiplicand, separately, by the multiplier, as in the former case; then these products, placed one after another, with their proper signs, will be the whole product required. multiplied by e, the product is either abc, acb, or bca, &c.; though it is usual, in this case, as well as in addition and subtraction, to put them according to their rank in the alphabet. It may here also be observed in conformity to the rule given above for the signs, that (+a)×(+b). or (-a)x(-b) =+ab; and (+a)×(-b), or (-a)×(+6) =-ab. CASE III. When both the factors are compound quantities. RULE.-Multiply every term of the inultiplicand separately, by each term of the multiplier, setting down the products one after another, with their proper signs; then add the several lines of products together, and their sum will be the whole product required. 1. Required the product of x2 - xy + y2 and x + у. Ans. x3 + y23. 2. Required the product of x3 + x2y + xy2 + y3 and x- y. Ans x - у. 3. Required the product of x2 + xy + y2 and x2 ху + y2. Ans. x + xy2 + y2. 4. Required the product of 3x2-2xy +5, and x2+2xy-3. Ans. 3x2 + 4x3y - 4x2y2 - 4x2 + 16ху - 15. 5. Required the product of 2a2 - 3ax + 4x2 and 5a2 - бах - 2x2. Ans. 10a - 27a3x + 34a2x2 -18ax3 - 8x. 6. Required the product of 5x3 + 4ax2 + 3a2x + a3, and 2x2 - 3ax + a2. Ans. 10x5 - 7ax2 - a2x3 - 3a3x2 +a5. 7. Required the product of 3x2 + 2x2y2 + 3y3 and 2x3 3x2y2 + 5y3. Ans. 6x - 5xy2 - 6x*y* + 21x2y + x2y + 15y. 8. Required the product of x3 - ax2 + bx c and x2 - dx + e. Ans. 25 ax - dx2+(b+ad+e) x2 - (c+bd+ae) x2 + (cd+eb) х — се. 9. * Required the product of the four following factors, viz. IV. (a + b), (a2 + ab + b2), (a - b), and (a2 - ab + b2). Ans. a - 6. 10. Required the product of a3+3a2x + 3ax2 + x3 and a3-3a2x + Зах2 - x3. Ans. a 3a2x2 + 3a2x2 - x. 11. Required the product of a + a2c + c2 and a2 - c2. Ans. a с. 12. Required the product of a2 + b2+c2 - ab - ас- be and a + b + c. Ans. a3-3abc + b3 + c2. DIVISION. DIVISION is the converse of mulplication, and is performed like that of numbers; the rule being usually divided into three cases; in each of which like signs give + in the quotient, and unlike signs -, as in finding their products.t It is here also to be observed, that powers and roots of the same quantity, are divided by subtracting the index of the divisor from that of the dividend. When the divisor and dividend are both simple quantities. RULE.-Set the dividend over the divisor, in the manner of a fraction, and reduce it to its simplest form, by cancelling the letters and figures that are common to each term. * I would advise the learner to perform the calculation of this example several ways; viz. First, by multiplying the product of the factors I. and II. by the product of the factors III. and IV. Secondly, by multiplying the product of the factors I. and III. by the product of the factors II. and IV. Thirdly, by multiplying the product of the factors I. and IV. by the product of the factors II. and III. The last method is the most concise. See Euler's Algebra, page 119, Vol. I.-ED. † According to the rule here given for signs, it follows that +ab -ab ab ==+==+=== b a, a, +ab as will readily appear by multiplying the quotient by the divisor; the signs of the product being then the same as would take place in the former rule. |