Therefore, 2a+3x is the greatest common measure. 3. What is the greatest common measure of 19ab2-1963=first remainder. Before dividing a2-5ab+4b2 by 19ab2-1963, we expunge from this last polynomial the factor 1962. Therefore, a-b is the greatest common measure. 4. What is the greatest common measure of -363+3ab2-a2b+a3 and 462-5ab+a2? Before dividing, we must multiply the polynomial -363 +3ab2-a2b+a3 by 4, in order that its left-hand term may be divisible by the the left-hand term of the other polynomial. (Art. 57.) FIRST OPERATION. -1263+12ab2- 4a2b+ 4a34b2-5ab+a2 -1263+15ab2-3a2b -3b-3a. Again, multiplying by 4 3ab2a2b+ 4a3 -12ab2-4a2b+16a3 -12ab2+15a2b- 3a3 -19a2b+19a3=first remainder. Before dividing 4b2-5ab+a2 by -19a2b+19a3, we expunge from this last polynomial, the factor 19a2, and then dividing, we have for the Therefore, b+a, or a−b, is the greatest common mea sure. 5. What is the greatest common measure of the two po15a5+10a4b+4a3b2+6a2b3-3ab4 12a3b2+38a2b3+16ab4-1065 lynomials { ดู Ans. 3a2+2ab-b2. 6. What is the greatest common measure of the two po ab+2a2-3b2-4bc-ac-c2 lynomials {at+2a25ab+4c2+8bc-12b2? Ans. 2a+3b+c, 7. What is the greatest common measure of x3-b2x and x2+2bx+b2? 8. What is the greatest common measure of a2-ab-262 and a2-3ab+2b2 ? Ans. x+b. Ans. a-2b. CASE II. (58.) To reduce a polynomial fraction, that is a fraction of which the numerator or denominator, or both, are polynomials, to its lowest terms, we have this RULE. Divide both numerator and denominator by their greatest common measure, found by Rule under Art. 52. EXAMPLES. 1. Reduce the fraction 36x6-18x5-27x1+9x3 to its simplest form. We see by a mere glance of the eye, that the numerator and denominator can both be divided by 9x3, by which di4x3-2x2-3x+1 vision the fraction becomes We must now seek the greatest common measure of 4x3-2x2-3x+1 and 3x2y2 -2xy2 —y2. Dividing the second of these by y2 (Art. 56), and multi plying the first by 3 (Art. 57), we have the FIRST OPERATION. 12x3-6x2- 9x+ 3|3x2-2x-1 12x3 —8x2 — 4x 4x+2 2x2-5x+3 Multiplying by 3 6x2-15x+9 6x2-4x-2 -11x+11=first remainder. 1 We must now repeat the operation upon 3x2-2x-1 and -11x+11. Dividing the second of these by 11 (Art. 56), we have for the x3 —xy2 2. Reduce to its lowest terms. In this example the greatest common measure of the numerator and denominator is found to be x+y. Hence, the fraction reduced becomes x2-xy (59.) To reduce a mixed quantity to the form of a frac tion. RULE. Multiply the entire part by the denominator of the fraction, to which product add the numerator, and under the result place the given denominator. EXAMPLES. 1. Reduce 11x++ to the form of a fraction. 7x In this example the entire part is 11x, which multiplied by the denominator 7x, gives 77x2, to which adding the numerator x+y, we have 77x2+x+y for the numerator of the fraction sought, under which placing the denominator 7x, we 77x2+x+y for the reduced form of 11x+ finally obtain x+y 7x 7x 2. Reduce x to the form of a fraction. |