without reduction, from the multiplication of a term of the divisor by a term of the quotient. Hence we may obtain a term of the quotient by dividing the term of the dividend affected with the lowest exponent of any letter, by the term of the divisor containing the lowest power of the same letter, and nothing prevents our operating upon the highest and lowest exponents of a certain letter alternately in the same example. (76.) From the examples of Art. 74, we perceive that a3-b' is divisible by a--b; and a'-b' is divisible by a-b. We shall find the same to hold true, whatever may be the value of the exponents of the two letters. That is, the difference of any two powers of the same degree is divisible by the difference of their roots. Thus, let us divide a"-b" by a-b. a°-b° la-b a°-a'ba a'b-b'. The first term of the quotient is a', and the first remainder is a*b—b3, which may be written b (a*—b1). Now if, after a division has been partially performed, the remainder is divisible by the divisor, it is obvious that the dividend is completely divisible by the divisor. But we have already found that a-b' is divisible by a-b; therefore a-b• is also divisible by a-b; and in the same manner it may be proved that ao—b is divisible by a-b, and so on. To exhibit this reasoning in a more general form, let us represent any exponent whatever by the letter n, and let us divide a"-b" by a-b. Dividing a" by a, we have, by the rule of exponents, a" for the quotient. Multiplying a-b by this quantity, and subtracting the product from the dividend, we have for the first remainder ba"--b", which may be written b (a"-b"). Now if this remainder is divisible by a-b, it is obvious that the dividend is divisible by a-b. That is to say, if the differ ence of the same powers of two quantities is divisible by then difference, the difference of the powers of the next higher degree is also divisible by that difference. Therefore, since a*b* is divisible by a—b, ao—b3 must be divisible by a-b; also. a-b, and so on. The quotients obtained by dividing the difference of the powers of two quantities by the difference of those quantities, follow a simple law. Thus, (a'-b')÷(a-b)=a+b. (a'-b3) ÷ (ab)=a2+ab+b2. (a'—b') ÷ (a—b)=a*+ab+a2b2+ab+bʻ. (a"-b")÷(a−b)=a"¬1+a"~b+a”—3b2+..+a2b′′3+ab+b”. The exponents of a decrease by unity, while those of b increase by unity. (77.) It may also be proved that the difference of two even powers of the same degree is divisible by the sum of their roots. Thus, (a2 —b2) ÷ (a+b)=a−b. (a*—b1) ÷ (a+b)=a3—a2b+ab3—b3. (a®—b®) ÷ (a+b)=a*—a*b+a3b3—a2b3+ab*—b3. Also, the sum of two odd powers of the same degres is divisıble by the sum of their roots Thus, (a3+b3)÷(a+b)=a2—ab+b2. (a+b)÷(a+b)=a*—a3b+a3ba—ab3+b*. (a'+b')÷(a+b)=a®—a'b+a*b*—a3b3+a2bˆ- ·a3°+·ba. (78.) The preceding principles will enable us to resolve va rious algebraic expressions into their factors. 1. Resolve as-b3 into its factors. Ans. (a2+ab+b3) (a− b) 2. Resolve a3+b3 into its factors. 7. Resolve 1+276' into its factors. 8. Resolve 8a3+2763 into its factors. (79.) One polynomial can not be divided by another polynomial containing a letter which is not found in the dividend; for it is impossible that one quantity multiplied by another which contains a certain letter, should give a product not containing that letter. A monomial is never divisible by a polynomial, because every polynomial multiplied by another quantity gives a product containing at least two terms not susceptible of reduction. Yet a binomial may be divided by a polynomial containing any number of terms. Thus, a-b' is divisible by a'+ab+ab2+b3, and gives for a quotient a-b. So, also, a binomial may be divided by a polynomial of a hundred terms, a thousand terms, or, indeed, any finite number. DIVISION BY DETACHED COEFFICIENTS. (80.) We have shown, in Art. 64, how multiplication may sometimes be conveniently performed by operating upon the coefficients alone. The same principle is applicable to aivision. Thus, take the example of Art. 73, to divide a2+2ab+b2 by a+b; we may proceed as follows: 1+2+11+1 1+1 1+1 1+1 The coefficients of the quotient are 1+1. Moreover, a'÷a =a; and therefore a is the first term of the quotient, and b the second. Ex. 2. Divide x-3ax3-8a2x2+18a'x-8a' by x2+2ax-2a 1-3-8+18-81+2-2 The coefficients of the quotient are 1-5+4, and it remains to supply the letters. Now x'x'=x2; and a'÷a'=a2. Hence x2, ax, and a2 are the literal parts of the terms, and therefore the quotient is x2-5ax+4a3. Ex. 3. Divide 6a-96 by 3a-6. Here, as we have the fourth power of a without the lower powers, we must supply the coefficients of the absent terms, as in multiplication, with zero. Ex. 4. Divide 3y'+3xy'-4x'y-4x' by x+y. Ans. 3y'-4x3. Ans. 2a-a'x. Ex. 5. Divide 8a-4a'x-2a'x2+a'x' by 4a2-x2. Ex. 6. Divide a'+4a'-8a-25a'+35a'+21a-28 by a'+ 5a+4. Ans. a'-a-7a'+14a-7. MISCELLANEOUS EXAMPLES. 1 Divide 3a-19a'b+25a2b2-25ab' by 3a-4ab+5b'. Ans. a-5ab. 2. Divide x*+2x2-4x3y2+16xy-15 by x2-2xy+5. Ans. x+2xy-3. 3. Divide a2x2-4a3bx+3acx+3a'b'+abc-10c' by ax-3ab +5c. Ans. ax-ab-2c. 4. Divide 20a*b*—22a3b'+11a2b3-3ab' by 4a3b3—2ab*+b3. Ans. 5a'b3-3ab1. 5. Divide x-xˆy2+2x2y*—y° by x3—x3y+y3. Ans. x3+x3y—y3. 6. Divide 3x + 15x*y' — 22ax1 — 2x1y3 — 10.x3y' + 14axy' — 5ax3y2+7a2x by x3+5xy3—7a. Ans. 3x-2xy' —ax. SECTION VI FRACTIONS. (81.) When a quotient is expressed as de cribed in Art. 16 by placing the divisor under the dividend with a line between them, it is called a fraction; the dividend is called the numerator, and the divisor the denominator of the fraction. Algebraic fractions do not differ essentially from arithmetica. tractions, and the same principles are applicable to both. The following principles form the basis of most of the operations upon fractions: 1. In order to multiply a fraction by any number, we must multiply its numerator, or divide its denominator by that number. nominator of the same fraction by a, we obtain also ab; that is, the original value of the fraction b has been multiplied by a. 2. In order to divide a fraction by any number, we must divide its numerator or multiply its denominator by that number. a2b Thus, the value of the fraction is ab. If we divide the a or b; and if we multiply the de nominator of the same fraction by a, we obtain is, the original value of the fraction ab has been divided by a. 3. The value of a fraction is not changed if we multiply or divide both numerator and denominator by the same number. |