ponent is equivalent to the reciprocal of the same quantity with an equal positive exponent. We also infer that a quantity may be transferred from the denominator to the numerator of a fraction, or the reverse, by changing the sign of its exponent. These conclusions are in accordance with the principle explained in Art. 6; for if a positive exponent indicates that a quantity is to be taken a certain number of times as a factor, an equal negative exponent should indicate that the quantity is to be taken the same number of times as a divisor. It will be shown hereafter that quantities having negative exponents can be operated on by the rules that are given for operating on quantities with positive exponents. This principle often enables us to change an indicated quotient to a simpler form without altering its value. Such a change is called reduction. 33. By reversing the rule for multiplying a polynomial by a monomial, we have the following rule for dividing a polynomial by a monomial: RULE. Divide each term of the polynomial by the monomial, and connect the quotients by their proper signs. 6. Divide 27am-18a2mb 5. Divide 14axy Yaby 13y by a3xys. — Ans. 2a-1x-2-a-2bx-3y — 13α-3 x-3y. 7. Divide 12a1(a + x)2 by 6a2(a + x)2. Ans. 18a3 (a + x)3 + 24a2(a + x)1 2a23a(a + x) + 4(a + x)2. Division of Polynomials. 34. To deduce a rule for dividing one polynomial by another, take the following example: The dividend and divisor are both arranged with reference to the same letter; the quotient of the first term of the dividend, by the first term of the divisor, is therefore the first term of the quotient. The product of the divisor by this term, subtracted from the dividend, gives a new dividend, which is treated in the same way, and so on to the end of the process. For convenience of multiplication, the divisor is written on the right of the dividend, and the quotient is written under the divisor. In all other respects, the operation is entirely similar to division in Arithmetic. Since all similar cases may be treated in the same manner, we have the following rule for the division of polynomials: RULE. I. Arrange both polynomials with reference to the same letter. II. Divide the first term of the dividend by the first term of the divisor, for the first term of the quotient. Multiply the divisor by this term, and subtract the product from the dividend. III. Divide the first term of the remainder by the first term of the divisor, for the second term of the quotient. Multiply the divisor by this term, and subtract the product from the first remainder, and so on. IV. Continue the operation, until a remainder is found equal to 0, or one whose first term is not divisible by that of the divisor. If a If a remainder is found equal to 0, the division is exact. remainder is found whose first term is not divisible by that of the divisor, without giving rise to fractions, the exact division is impossible. In that case, write the last remainder over the divisor and add the result to the quotient already found. Here the quotient is fractional, and the division is not exact. In this example, the operation does not terminate, but may be continued to any desired extent. EXAMPLES. 1. Divide a2 + 4ax + 4x2 by a +2x.. 3. Divide a3 + 5a2x + 5αx2 + x3 by a+ x. Ans. at 4ax + x2. 4. Divide a1 — 4a3y + 6a3y2 — 4ay3 + y1y a2—2ay + y2. 5. Divide a1 b4 by a3 + a3b + ab2 + bì Ans. a2-2ay + y2. Ans. 4x+8x2 + 1 + 32. 7. Divide 26 3x4y2 + 3x2y1 — y by Ans. x+3x2y + 3xy + y3. 8. Divide x + x2ny2n + y1n by n x2n + x^ yn + Zn. Ans. x2-x"y" + y2n. i |