It may not be amiss to observe, that when any quantity is divided by itself, the quotient will be unity, or I; because any thing contains itself once: thus xx gives 1, and 2ab divided by ✓ 2ab gives 1. NOTE 1. To divide any power by another of the same root; subtract the exponent of the divisor from that of the dividend, and the remainder will be the exponent of the quotient. Thus, 3. When one term is + and the other -; the quotient must be -, because + in the divisor X in the quotient produces in the dividend; or - in the divisor X + in the quotient gives - in the dividend. So that the rule is general; like signs give +, and unlike signs give, in the quotient. Thus, the quotient of a divided by a3 is a3-3, , or as. n-r And that of x" by x" is xr. But it is to be observed, that when the exponent of the divisor is greater than that of the dividend, the quotient will have a negative exponent. 2 Thus, the quotient of xs divided by x7 is x5-7, or *-* And that of ax* by x5 is ax-3. 2 And these quotients, viz. & and ax-3, are respective axn In like manner, ax" divided by cx " gives cx m 2 And the quotient of a2+x* divided by a2+x* is a2+x21. Moreover, a divided by a gives a 5 3-1 a+x divided by a+xl gives a+x19 a+x=a+ m n 2 = m-r And ab+x*" divided by ab+x*\" gives ab+x*" SCHOLIUM. SCHOLIUM. T When fractional exponents of the powers of the same root have not the same denominator, they may be brought to a common denominator, like vulgar fractions, and then their numerators may be added, or subtracted, as before. Thus, the quotient of ac+x divided by ac+x14 is NOTE 2. Surd quantities under the same radical sign are divided, one by the other, like rational quantities, only the quotient, if it do not become rational, must stand under the same radical sign. When the divisor is a simple quantity and the dividend a com pound quantity. RULE. Divide every term of the dividend by the divisor, as in the first case. NN EXAMPLES. When the divisor and dividend are both compound quantities. RULE. 1. Range the terms according to the powers of some letter in both of them, placing the highest power of it first, and the rest in order. 2. Divide the first term of the dividend by the first term of the divisor, and place the result in the quotient. 3. Multiply the whole divisor by the quotient term, and subtract the product from the dividend. 4. To the remainder bring down as many terms of the dividend as are requisite for the next operation; call the sum a dividual, and divide as before; and so on, as in Arithmetic. EXAMPLES. EXAMPLES. 1. Let it be required to divide a3-3ax by a-tx. a+x)a3. a3+a*x x-3x2+x * The process may be explained thus : First, a3 divided by a gives a for the first term of the quotient, by which we multiply the whole divisor, viz. a+x, and the product is a3+a2x, which, being taken from the two first terms of the dividend, leaves -4a2x ; to this remainder we bring down -3ax, the next term of the dividend, and the sum is -4a2x-3ax2, the first dividual; now dividing -4a2x, the first term of this dividual, by a, the first term of the divisor, there comes out -4ax, a negative quantity, which we also put in the quotient; and the whole divisor being multiplied by it, the prod. uct is -4a2x-4ax2, which being taken from the first dividual, the remainder is + ax2; to which we bring down x3, the last term of the dividend, and the sum is + ax2+x3, the second dividual; and + ax2, the first term of the second dividual, divided by a, the first term of the divisor, gives x2 for the last term of the quotient; by which we multiply the whole divisor, and the product is tax2+x3, which being taken from the second dividual leaves nothing; and the quotient required is a2-4ax+x2. 2 3 2 |