DEGREE OF PRODUCTS. 333 EXERCISES. 10. What is the sum of fifteen times x and seventeen times ? 15x + 17x how many times x ? = 20. ax + bx= ? 3°. 2(a+b)+3(a+b)=? 4°. a(c+d) +b(c+d)=? 5°. 106 76=? 6°. 35b - 20b = ? 7°. ac - bc =? 8°. ad+bd=-(ad - bd)=? 9°. mx+nx-rx = ? 10°. ac-ad+bd — bc = (a - b) (c - d,), how? 8 11°. Resolve a2 — b2, a1 — b1, ao — b3, ao — bo, a1o —b1o, a2b2, into their simplest factors. 2n 12°. Resolve x2 + 2x + 1, x2+1 −2x, 1-2(xy −x+y)+xx +yy, into their prime factors. PROPOSITION X. The degree of a product is equal to the sum of the num- (12) bers indicating severally the degrees of its factors. For, as two factors, ab, multiplied by a third, a,, give a product of three factors, aba,, so a monomial of p factors abc ... [p], multiplied by an additional factor, a,, gives a product of p+1 factors, abc... Xa2 [p+1], and by a second factor, abc ...Xab2 [p+2]; and generally, abc... [p] Xa2b2c2 ...[9] = abc... Xaqb2C2 ... [P+q]; abc... [p] Xab.c2 ... [q] × а3b3c3 ... [r] = abc ... × ab2c2 ... X Xazb3c3... [p+q+r]; 2 ... in general, abc ... [p] × ab2c2 ... [9] × α3b3C3 ... [r] × ... = abc ...abc... a ̧b3c3 ... [p+q+r+...]. (12) Cor. 1. Powers of the same letter are multiplied together (13) by adding their exponents. For from (12), making b, c,... A29b2;C2, ... A3,b39C39 ... all = a, there results, or ... aaa ... [p] × aaa [9] × aaa... [r] x ...aaa... a2 • a • a • (13) Cor. 2. A quantity is involved by multiplying its exponent (14) by the index of the power to which it is to be raised. For, making p, q, r, ... all = m and n in number (13), becomes Cor. 3. Powers of the same letter are divided by diminish (15) ing the exponent of the dividend by that of the divisor. Cor. 4. When one polynomial is divided by another, the (16) highest power of any letter in the dividend divided by the highest power of the same letter in the divisor, gives the highest power of that letter in the quotient. For (15), divisor a quotient ao = dividend a2+?. Scholium. It will be found convenient in division to arrange (3) the polynomials according to the descending powers of a given letter. EXERCISES. 1o. Divide a b by a a-b. 2o. Divide a2 — b2 by a-b. 3°. Divide a3-b3 by a-b. 4°. Divide a b1 by a-b. 5o. Divide a b3, ao — bo, a2 — b2, ..., a” — b”, by a — b. PROPOSITION XI. 1o. Multiplying the dividend while the divisor remains (17) the same, or dividing the divisor while the dividend remains the same, multiplies the quotient; 2o. Dividing the dividend while the divisor remains the same, or multiplying the divisor while the dividend remains the same, divides the quotient; .. MULTIPLICATION OF FRACTIONS, 35 3°. The value of the quotient is not altered by multiplying or dividing both divisor and dividend by the same quantity. Def. 7. A Fraction is an expression of division, and arises from an impossibility of performing the operation. Therefore, indicating the dividend, now called the numerator, by N, the divisor or denominator by D, and the quotient or value of the fraction by V, we have N: D= V, or N = DV; hence (17), Cor. 1. A fraction is multiplied by multiplying its numera- (18) tor, N, or by dividing its denominator, D, [N • R = D • VR, &c.]. Cor. 2. A fraction is divided by dividing its numerator, or (19) by multiplying its denominator. Cor. 3. A fraction is changed in form, without being (20) changed in value, by multiplying or dividing both numerator and denominator by the same quantity. Cor. 4. If a fraction be multiplied by its denominator, the (21) product will be the numerator. For (18) the fraction multiplied by d: nd d = (nd): d = n. Cor. 5. Fractions are multiplied together by taking the (22) product of their numerators for a new numerator, and that of their denominators for a new denominator. multiplying both numerator and denominator by D, (20) ; Cor. 6. A fraction may be raised to any power by involving (23) its numerator and denominator separately, and consequently, any root of a fraction will be found by an evolution operated upon its terms. For, making N2, N3, ... = N, == 39... = [m] = D, (22) becomes NNN... [m] DDD ... [m]' or : Cor. 7. To divide by a fraction, invert it, and proceed as (24) in multiplication. N Let Qbe any quantity divided by a fraction; multiplying D both dividend and divisor by (17), we have N N N D : 1 Cor. 8. The reciprocal of a fraction is the fraction inverted. (25) Cor. 9. Any quantity, whether whole or fractional, will be (26) reduced to a given denominator, by being multiplied by this denominator and set over it. least common denominator, M, must be divisible by each of the denominators, D, D2, D3, ... of the given fractions, or, must be the least common multiple of all the denominators, if we would have the resulting fractions freed from subdenominators, .. Cor. 10. To add or subtract fractions, find the least com- (27) mon multiple of the denominators, to which, as a common denominator, reduce all the fractions, then add or subtract the numerators (55). This rule will frequently be superseded by the following: Cor. 11. To free a fraction of subdenominators, multiply (28) its terms by the least common multiple of the subdenominators (20). where, if we Thus, would make the subdenominators b, d, g, i, k disappear, M must be divisible by each. Scholium I. The rule for managing the signs has already been given in (6.), but it may be well to observe that, of the three signs pertaining to a fraction-viz., the sign before the fraction and those before its terms-an even number produces plus, an odd number minus, and any two may be changed at a time. Thus, + - + =+ (+ : +) =+ (+) = +, ++=+(− : +)=+(−)=−, + + +==+(− : −)=+(+) = +,+==+ (+ : −) =+ (−) =—; ·(+ '+ = (-) — = (+ : −) —=—— ·−=(+)− = (+ : |—'— = (+) − = (− : −)-== -==- (+ :-) = ~(~) = +. Scholium II Whenever it may be foreseen that, by performing an operation the same factor would be introduced into the numerator and denominator, it should be suppressed. Scholium III. Additions and subtractions will frequently be better performed in part before reducing to a common denominator. Def. 8. An Equation is an algebraical expression consisting of two members, separated by the sign of equality [=]. Equations are of different kinds. 1o. An identical equation is one in which the members are the same, as a = a. 2o. An equation of operation has one of its members derived from the other of these we have already had many examples, such nc = (m — n)c. as mc 30. An equation of condition expresses a determinate relation that must exist between certain quantities, not distinguished as known and unknown, as (a+b) = p-q. 4°. The word equation more commonly indicates a relation between known and unknown quantities, such that the latter may be |