(3) (a−b)√x + (m−n)√y + √2 (4) (m+n) y3-(ab)x+ary (n−p) y2—(2a+ b)x2—bxy (p-2nly-(c-3a)x2+czy (q-m)y-(c+2d)x2-dry (5) Add a x2+by+c to dx2+hy+k. (6) Add together x2 + x y + y2; a x2—a xy +ay2; and — by2+bxy+b x2 x2+ xy + y2 x2 — xy + y2. and 2 2 (8) What is the sum of (a+b) x + (c–d) y x√2; (a - b) x + (8c+2d)y+5x√2; 2 bx +3 dy-2x/2; and -3bx-dy-4x√2. (5) (a + d) x2 + (b + h) y + c + k. (6) (1+a+b) x2 + (1 − a + b) xy + (1 + a−b) y'. 6. THE subtraction of monomials is indicated by placing the sign -- between the quantity to be subtracted and that from which it is to be taken. Thus a-b signifies that the quantity denoted by b is to be subtracted from that denoted by a; and if 2 x y is to be subtracted from a2 + y2, the result is represented by a2 + y2 — 2 xy. Place the quantity to be subtracted under that from which it is to be taken; change the signs of all the terms in the lower line from + to and from to +, or else conceive them to be changed, and then proceed as directed in Addition. It is evident, that if all the terms of the quantity to be subtracted are affected with the sign+, we must take away, in succession, all the parts or terms of the quantity to be subtracted; and this is indicated by affecting all - its terms with the sign Also, if cd is to be subtracted from a + b, then c taken from a + b is expressed by a + b. c; but if c-d, which is less than c by the quantity d, be taken from a + b, the former difference, a + b − c, will obviously be too small, and will require the addition of d to make up the deficiency; and therefore c - d taken from a + b is expressed by a + b − c+d, which is equivalent to the addition of -c+d to a + b. Hence the reason for the change of the signs in the quantity to be subtracted. Or thus: Since c―d is to be subtracted from a +b; then, if c be subtracted, we shall have subtracted too much by d; hence the remainder a + b − c is too small by d; and therefore, to make up the defect, the quantity d must be 2a+ax+x3-12a'x+20ax3- 4x3 +6a2x2-10ax3 (11) 4y2-4yx+x2-2a(x+y)+6√/a2x2-8" /b2 —y 7. In order to indicate the subtraction of a polynomial, without actually performing the operation, we have simply to enclose the polynomial to be subtracted within brackets or parentheses, and prefix the sign. Thus, 2a3 —3a2b+4ab2—(a3+b3+ab2) signifies that the quantity a3+b3+ab2 is to be subtracted from 2a3-3a2b+4ab2. When the operation is actually performed, we have by the rule 2a3-3ab+4ab2—(a3+b3+ab2)=2a3-3a2b+4ab3—a3—b3—ab2 = a3-3a2b+3ab2—b3. 8. According to this principle, we may make polynomials undergo several transformations, which are of great utility in various algebraic calculations. Thus, a3-3a2b+3ab2-b3—a3-(3a2b-Sab2+b3) =a3—b3—(3a3b—3ab3) =a3+3ab2-(3ab+b3) = −(—a3+3a2b—3a2b+b3) And x2-2xy+y2=x2—(2xy—y3)=y3—(2xy—x2). (3) From m2 n2x2—2 m n p q x+p2 q2 take p2 q2x2-2 p q m n x + m2 n3. (4) From a (x+y)—bxy+c(x-y) take 4(x+y)+(a+b) x y−7 (x—y). (5) From (a+b)(x+y)—(c–d) (x−y)+h2 take (a—b) (x+y)+(c+d) (x-y)+k2. (6) From (2a-5b)√x+y+(a−b) x y-cz take 3 b x y−(5+c) z1— (3 a—b) (x+y)*. (7) From 2x-y+(y-2x)-(x-2y) take y-2x-(2y-x)+(x+2y). (8) To what is a+b+c−(a−b)—(b—c)—(—b) equal ? ANSWERS. (3) (m2 n2—p2 q2) x2 +p2 q2—m2 n2 or (m2 n2—p2 q2) x2—{m2 n2—p2 q2). (4) (a—4)(x+y)−(a+2b) xy+(c+7)(x−y). (5) 2b (x+y)-2c (x-y)+h2-k2. (6) (5a—6b)√x+y+(n−4b)xy+5z2. (7) y---x'. (8) 2b+2c. 93 MULTIPLICATION. 9. MULTIPLICATION is usually divided into three cases: (1) When both multiplicand and multiplier are simple quantities. (2) When the multiplicand is a compound, and the multiplier a simple uantity. (3) When both multiplicand and multiplier are compound quantities. CASE I. 10. When both multiplicand and multiplier are simple quantities. To the product of the coefficients affix that of the letters. Thus, to multiply 5 a x by 4 axy, we have 5 × 4 = 20; a x × axy — a2x2 y; .. 5ax × 4axy = 20 × a2x2y = 20 a2x2y = product. RULE OF SIGNS IN MULTIPLICATION.* The product of quantities with like signs, is affected with the sign +; the product of quantities with unlike signs, is affected with the sign or multiplied by + and multiplied by - give +; +multiplied by and multiplied by + give -; or like signs produce + and unlike signs The truth of this may be shown in the following manner:— (1) Let it be required to multiply + a by + b. Here a is to be taken as often as there are units in b, and the sum of any number of quantities affected with the sign +, being +, the product ab must be affected with the sign+, and is therefore +ab. (2) Multiply + a by — b, or — a by + b. In the former case -b is to be taken as often as there are units in a, and in the latter a is to be taken as often as there are units in b; but the sum of any number of quantities affected with the sign is also; hence in either case the product ab must be affected with the sign —, and is therefore ab. • Let N represent either a number or any quantity whatever, and put a = + N ; b = — N Then, since a = + a, and b = + b, we shall have +a+N; + b = — N a=N; b = + N. Now, if in these four last equations we substitute the values of a and b from the first two eqnations, we have + (+N) = + N; + (−N) = — N — (+N) N; — (— N) = + N. Now, in each of these formulas, the sign of the second number is what is named the product of the two signs of the first number; hence the truth of the rule of signs. Thus instead of writing a × a × a × a × a, or a a a a a, which represents five a's multiplied together, we write a3, where 5 is called the exponent or index of a. Similarly b × b × b x b x b x b x b x b x b × b, or b.b.b. b. b. b. b. b. b. b, or b b b b b bbb bb; or the continued product of 10 b's is written more briefly b1o, where 10 is the exponent or index of b. The exponent or index of a number is, therefore, a number written a little above a letter to the right, and denotes the number of times which the number When no exdesignated by the letter enters as a factor into a product. ponent is expressed, the exponent 1 is always understood; thus a' and a signify the same thing. The products thus formed by the successive multiplication of the same number by itself, are in general called the powers of that number. Thus a is the first power of a; a × a = aaa2 is the second power of a, or the square of a; aaa = a3 is the third power, or cube of a; aa aa a = a3 is the fifth power of a, and a aaa..... to n factors = a", is the nth power of a, or the power of a designated by the number n, X. The square root of any expression is that quantity which, when multiplied by itself will produce the proposed expression, and, in numbers, is generally denoted by the symbol, which is called the radical sign. Thus the square root of 9 is 93, and √/a2 = a, is the square root of a2; for in the former case 3 x 39, and in the latter a × a = a2. XI. The cube root of any expression is that quantity which, when multiplied twice by itself, will produce the proposed expression. The fourth, or biquadrate root of any expression is that quantity which, when multiplied three times by itself, produces the given expression; and the nth root of any expression is that quantity which, multiplied (n-1) times by itself, produces the proposed expression. Thus the cube root of 8 is 2; for 2 × 2 × 2 = 8, the fourth root of a1 is a; for a. a. a. a = a, and the nth root of 2a y" is 2 for y; .... to n factors = x xy xxy xxy.. Xy.y.y.y. to n factorsx" y". .... x. x.... to n factors The roots of expressions are frequently designated by fractional or decimal exponents, the figure in the numerator of the fractional exponent denoting the power to which the expression is to be raised or involved, and the figure in the denominator denoting the root to be extracted or evolved. Thus the symbol of operation for the square root of a is either a or a1; for the cube root it is a, or a; for the fourth root Va, or a1; and √/a, or denotes the nth root of a. Also a3, or a, denotes the sixth root of the fifth power of a; and a", or "\/aTM, signifies the nth root of the mth power of a. XII. A rational quantity is that which has no radical sign, or fractional exponent annexed to it, as 3 m n, or 5x2 y2. XIII. An irrational quantity is that which has no exact root, and is expressed by means of the radical sign, or a fractional exponent, as √2 Va2, or x+y+. XIV. The reciprocal of any quantity is unity divided by that quantity; thus the reciprocals of a2, 2, 5, 2, are respectively a''; but the following notation is generally used, as being more commodious: thus the fractions are expressed by a ̄2, x3, ya3, 273. |