EXAMPLES. 1. What is the quotient of 2ab+b3+2ab2+a3 divi-, ded by a2+b2+ab ?. Arranging the terms according to the powers of a, and proceeding agreeably to the above rule, we have this OPERATION. Dividend q3+2a2b+2ab2+b3\a2+ab+b2=divisor. 2. What is the quotient of 4x3+4x2-29x+23 by In this example, we find 2 for remainder, which, being placed over the divisor 2x-3, gives the fraction 3. Divide a1+a2z2+z1 by a2+az+z2. 4. Divide a3-3a2b+3ab2-b3 by a-b. Ans. a2-2ab+b2. 5. Divide 25a2-10ax+x2 by 5a-x. 6. Divide 6a-96 by 3a-6. Ans. 5a-x. Ans. 2a3+4a2+8a+16. 7. Divide 3x26x3y 14xy + 37xy by 3x2 — 3x2 -5xy+2y2. 8. Divide x2+2xy + y2 by x+y.. Ans. x2-7xy. Ans. x+y. 9. Divide x 4x3y + 6x2y2-4xy3 + y1 by x2 -2xy+y2. Ans. x2-2xy+y2. Ans. 8m2n3-5mn*. 10. Divide 64m*n*-25m2n3 by 8m2n3+5mn*. CHAPTER II. ALGEBRAIC FRACTIONS. (41.) In our operations upon algebraic fractions, we shall follow the corresponding operations upon numerical fractions, so far as the nature of the subject will , admit. CASE I. To reduce a monomial fraction to its lowest term, we have this RULE. I. Find the greatest common measure of the coefficients of the numerator and denominator. (See Arithmetic.) II. Then, to this greatest common measure, annex the letters which are common to both numerator and denominator, give to these letters the lowest exponent which they have, whether in the numerator or denominator. The result will be the greatest common measure of both numerator and denominator. III. Divide both numerator and denominator by this greatest common measure, (by rule under Art. 36,) and the resulting fraction will be in the lowest terms. 1. Reduce 375a3bxy EXAMPLES. to its lowest terms. The greatest common measure of 375 and 15 is 15, to which annexing abxy, we have 15abxy for the greatest common measure of both numerator and denominator. Dividing the numerator by 15abxy, we find 375a3bxy-15abxy=25a2. In the same way we find 15ab2xy315abxy=by2, hence, we have 375a3bxy 25a2 15ab xy3 by which, by Rule under Art. 36, becomes In this example, the greatest common measure of the numerator and denominator is 7xyz3, hence, dividing both numerator and denominator of our fraction by 7xyz3, we find (42.) To obtain a general rule for reducing a fraction whose numerator, or denominator, or both, are polynomials, it would be necessary to show how to find the greatest common measure of two polynomials, a process which is too complicated and difficult for this place. There are, however, many polynomial fractions, of which, the common measure of their respective numerators and denominators are at once obvious, and of course they are then readily reduced. We will illustrate this by a few 1. Reduce 3xy+xz EXAMPLES. to its simplest form. |