I 1 17. Add 4ax 130 + 3x2, 5x2 + 3ax -+ 9x, 7xY 4x2 + 90, and V x + 40 - 6x2 together. Ans. 7ax + 8x® + 7xy. 8. Add 2a" 3ab + 263 3a, 373 503, 403 253 +- 5ab + 100, and 20ab + 16a2 be 80 together. Ans. 13a + 22ab + 383 as C3 + 20 - bc. 5a 30 7 bc 8a 724 9. Add + 9 ( b d 6 ab + x -) and + a a V bc +6,66 , *, together. ab + x 12 d ab + x Ans. 十 5 3 b d a 10. Add 3a2 + 4bc 62 +10, 5a2 + 6bc + 2e2 and -- 4a2 - 9bc 10e + 21 together. Ans. bc - 6am - 9e + 16. 15, SUBTRACTION. SUB'TRACTION is the taking of one quantity from another; or the method of finding the difference between any two quantities of the same kind; which is performed as follows: RULE.—Change all the signs (+ and —- ) of the lower line, or quantities that are to be subtracted, into the contrary signs, or rather conceive them to be so changed, and then collect the terms together, as in the several cases of addition. EXAMPLES. 5a? 26 2a2 + 5b 202 2y + 3 4x2 +9y - 5 . 5xy + 8x 2 3xy 8x --- 7 3a2 - 76 3х2 lly + 8 2xy + 16x + 5 . . * This rule being the reverse of addition, the method of operation must be so likewise. It depends upon this principle, that to subtract an affirmative quantity from an affirmative, is the same as to add a negative quantity to an affirmative. Thus, according to Laplace, we can write a=a-tb-6 (1), a. --C=a-otb-b e); so that if from a we are to subtract t-b, or -- b; or what amounts to the same thing, if in a we suppress t-b, or b; the remainder from transformation (1), must be a - b in the first case, and a to in the second. Also, if from ac we take away +b, or -- b, the remainder, from (2), will be a--0-b, or ac b, or a-c+b.-ED. + - 5xy - 8a 4 Vax -20y 5x + 7 x -44 + 3xy-77 3 V ax 6.22 8.00 - 8x®y-8a+75 Vax-2x+y + 5aya –+ 8x +2yx--4y EXAMPLES FOR PRACTICE. 7 ax 1. Find the difference of } (a + b) and ž (a - b). Ans. b. 2. From 3x – 2a -b +7, take 8 b+7, take 8 - 3b tat- 4x. 3a - 1. 3. From 3a + b + c m 2d, take b 8c + 2d -8. Ans. 3a +9c - 40 +8. 4. From 13x2 - 2ax + 96%, take 5x2 62. Ans. 8x2 + 5ax + 1064. 5. From 20ax - 5.7 2. + 3a, take 4ax + 5x2 Ans. 16ax 10 V x + 4a. 6. From 5ab +262 - c+bc - b, take 72 — 2ab + bc. Ans. 7ab + 32 -0-b. 7. From ax3 d, take bx2 + ex 2d. Ans. axi - 26x + (c-e) x + d. 8. From - 6a - 46 - 12c + 13x, take 4x -- 9a + 4b - 50. Ans. 3a + 9x - 86 86 – 76, 9. From 6x*y - 3 (xy) — 6ay, take 3x*y + 3 (xy) Ans. 3x2y - 6V (æy) - 2ay. 10. Frorn the sum of 4ax 150 + 4x2, 5x2 + 39x + 10x3, and 90 - 2ax 12 V (x); take the sum of 2ax - 80 + 7%, I 7x2 8ax - 70, and 30 - 4V (x) — 202 + 4aRx2. Ans. 11ax + 60.- 32 4à®202, I 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 multiply. ing, 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, axa’, or al Xa? = d?; Q2 x <= qb; as xaš = qő; and an X an amt n. 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 v (a + b). 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. * 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 d, it is plain that the product, in this case, must be less than ab, because 8 -- 6 is less than B; and, consequently, when each of the terms of the former are multiplied by a, as above, the result will be (B-6) Xa= AB ab. For if it were as tab, the product would be greater than ab, which is absurd. Also, if u be greater than b, and a greater than a, and it is required to multiply B-5 by A - a, the result will be (B-b) X(1 — a) = AB — AB — bA tab. For the product of B- b by A is A ( Bb), or AB-- Ab, and that of b by - a, which is to be taken from the former, is - a (Bb) 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 ab; and consequently since the first part of this product is AB, the second part must be + ab; for if it were - ab, a greater part than ab would be to be taken from A (B--b), 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 B 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 ', 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 (t-a) X(+6): or (--0) X(--1)= tab; and (ta) X(-), or(-a)x1+67 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 2* — xy + y2 and x + y. Ans. 23+73. 2. Required the product of x3 + xy + xy+ ys and x - y. Ans 24 - 44 3. Required the product of 22 + xy + y' and 22 - 2y + y2. Ans. 204 + xcoy2 + y4. 4. Required the product of 3x2 – 2xy +5, and x2 + 2xy - 3. Ans. 3x+ +- 4xy -- 4xya — 4x2 + 16xy — 15. 5. Required the product of 2a2 3ax + 4202 and 5a 6ax - 28.2. Ans. 10a* - 27a’x + 34aRx2 -- 18ax 80c4. 6. Required the product of 5x3 + 4ax2 + 3aRx + a', and 2002 Зах +а”. Ans. 10x5 7ax4 a 203 3a%xta. 7. Required the product of 3x2 + 2xy + 3y and 2003 – 3x+ya + 5y. Ans. 6x6 – 5x®y -- 6x*y* + 21x®y3 + xy + 15yo. 8. Required the product of ** -- ax® + bx - c and x2 - dx +e. Ans. 26 Fax+ - dx4 + (6 + ad + e) *3- (c + bd +- ae) 2012 + (cd + eb) ce. |