In this example, the numerator cannot be divided by the divisor. But we have seen that a fraction is multiplied by dividing the denominator; on the other hand, a fraction is divided by multiplying the denominator. m The fraction means that m is divided into n equal parts; and if the denominator be multiplied by 5, for example, m will be divided into 5 times as many parts as it was before; consequently, the parts will be as great as -before, that is, of is In like manner, if the de m m m 5 n nominator of be multiplied by a, m will be divided into n a times as many parts as it was before, and the parts will Hence, to divide a fraction by an integral quantity, multiply the denominator by the integral quantity. ART. 58. Combining this rule with the preceding, we have the following GENERAL RULE FOR DIVIDING A FRACTION BY AN INTE GRAL QUANTITY. Divide the numerator, if it can be done, if not, multiply the denominator, by the integral quantity. FACTORS, OR DIVISORS OF ALGEBRAIC QUANTITIES. ART. 59. A prime quantity is one that can be divided by no entire and rational quantity, except itself and unity. Thus, a, b, and a+m are prime quantities. Two quantities are prime with regard to each other, when no quantity, except unity, will divide them both without a remainder. Thus, ab and c d, although neither of them is a prime quantity, are prime with regard to each other. Remark. Although, algebraically considered, we call a, b, and c, prime quantities, they are strictly speaking such, only when they represent prime numbers. We have frequent occasion to separate quantities into their prime factors. In a monomial, this operation is attended with no difficulty. We have only to find, according to the method usually given in arithmetic, the prime factors of the coefficient, and to represent them as multiplied together and followed by the several letters, each written as many times as it is a factor. Thus, 18 a2 m3 3.3.2aammm. In this example, the dif ferent prime factors are 3, 2, a and m; 3 is contained twice, 2 once, a twice, and m three times, as a factor. The same quantity may be expressed in its factors thus: 2.32 a2 m3, in which the exponents show how many times each quantity enters as a factor. When a quantity is the product of a monomial and a prime polynomial, in order to separate it into factors, it is only necessary to divide it by the greatest monomial that will exactly divide all the terms, and to place the divisor, separated into prime factors, before the quotient, the latter being included in a parenthesis. Thus, am+ana (m+n), in which the factors are a and m+n. In like manner, 50 a2 b2 +25 a b3 = 5o a b2 X (2 a+b), the factors of which are 5, a, b, and 2 a+b. Let the learner separate the following quantities into prime factors. 1. 16 a4 b. 2. 40 x2 y2. 3. 120 m y. 4. 15 x2 y3. 5. 225 my. 6. 54 x3 y2. 7. 3ax+7ay. Ans. a (3x+7y). 8. 2 a3+6a2 m. 9. 25 m3 -5 m2 p 10 54 a3 b2 c. -27 a b2 c2. 11. 81 m2 xy +27 m2 pq-5 m3 y. ART. 60. When a quantity is the product of several polynomials, the process of finding its factors becomes more difficult; but in many cases some of the factors may be easily ascertained. 1. Any power of a polynomial may evidently be separated into as many factors, each equal to that polynomial, as there are units in the exponent of the power. Thus, x2+2xy + y2 = (x + y)2 = (x+y) (x+y); and x3+ 3x2y + 3 x y2+y3=(x+y)3=(x+y)(x+y)(x+y). 2. The difference between the second powers of two quantities can be separated into two factors, one of which is the sum and the other the difference of those quantities. Thus, x2- y2 = (x + y) (x —y); also, x1—y1 = (x2 + y2) (x2 — y2) = (x2 + y2) (x + y) (x − y). 3. The difference between similar powers of two quantities can be separated into at least two factors, one of which is the difference of those quantities. Thus, x— -y, x2 — y2, x3 — y3, &c., are each divisible by x- -y. 4. The difference between similar even powers of two quantities, the powers being above the second, can always be separated into at least three factors, one of which is the sum, and another the difference, of the quantities. Thus, m1— n1 (m2 + n2) (m2 — n2) = (m2 + n2) (m+n) (m—n). 5. The sum of similar odd powers of two quantities can be separated into two factors, one of which is the sum of the quantities. Thus, x3 + y3 = (x + y) (x2 — x y + y2). The quantity x6 — y6 can be separated into four factors. Thus, x6 — y6 — (x3 + y3) (x3 — y3) = (x + y) (x2 — x y + y2) (x —y) (x2 + xy + y2). = Let the learner separate the following quantities into prime factors. 1. 6a2-6 b2. Ans. 2.3(a+b) (a—b). 2. 3x2-6xy+3y2. Ans. 3(x-y) (xy). 3. x2+2x+1. 4. 4x2+8x+4. 5. x4-y1. 6.9 x3 +9 y3. 9. 12m5+12 n5. 10. 623-18x2y+18 x y2-6 y3. Remark. Any power, also any root, of 1 is 1. 11. 9 m2 +9. 12. 18 x8-18. SECTION XVII. SIMPLIFICATION OF FRACTIONS. ART. 61. Both numerator and denominator of a fraction may be multiplied by the same quantity, without changing the value of the fraction; for multiplying the numerator multiplies the fraction, and multiplying the denominator divides the fraction; but when a quantity is multiplied, and the result is divided by the multiplier, the value of that quantity remains unchanged, |