1296+360y+25y3 72y2 = 36+5y 15. C. F. 1296y2 +360y3+25y4-9y4-2592y2 +360y3 16. Transposing and uniting, 16y4=1296y2 17. Extracting the square root, 4y2=36y 18. Dividing by y, 4y=36, and y=9 19. Substituting y's value in 11th, 12x+15x=81, & x=3. Therefore 22-y, x, y, y2÷x, =1, 3, 9, 27. DIVISION BY COMPOUND DIVISORS. Art. 462, Example 6. a+1)a3+a2+a3b+ab+3ac+3c a2b+ab a2b+ab 3ac+3c 3ac+3c a2+ab+3c=the quotient. Example 7, a+b—c)a+b-c-ax-bx+cx 1- the quotient. a+b-c Example 8. 2a2—ax+x2)2a1 —13a3x+11a2x2-8ax3+2x4 2a4a3x+a2x2 -12a3x+10a2x2-8ax3 -12a3x+6a2x2-6αx3 4a2x2-2ax3+2x1 4a2x2-2ax3+2x+ a2-6x+2x2=the quotient. Example 11. 3a+y)6ax+2xy-3ab-by+3ac+cy+h 2x-b+c+h÷(3a+y)=the quotient. Example 12. b-3)a2b-3a2+2ab-6a-4b+22 a2b-3a2 2ab-6a 2ab-6a a2+2a-4+10÷(6-3)=the quotient. Example 14. a+√y)a+√y+ar √y+ry a+ √y x2-2ax+a3 the quotient. Example 16. y-8)2y3-19y3+26y-17 2y3-16y2 -46+22 -46+12 10 Rem. X 1 x5+x2+x3+x2+x+1=quotient. Example 18. 2x2 +3x−1)4x1 — 9x2+6x−3 4x4+6x3-2x2 -6x3-7x2+6x -6x3 -9x2+3x 2x2+3x-3 2x2+3x-1 2x2-3x+1-2÷(2x2+3x-1)=quotient. -2 Rem Example 19. a+2b)a1+4a3b+3b1 a4+2a3b -2a3b+4a2b a3-2a3b+4ab2+4a3b-8b2-863-quot. -8ab2+16b4 Example 20. x2 −ɑx+a2)x4 a2x2+2a3x-a1 x2+ax-a2=the quotient. x4-ax3 + a2x2 ax3-2a2x2+2μ3x 3b4 Rem. —a2x2+a3x— a1 +b being the last divisor is there- -bx-b2 fore the greatest common measure. —bx-b2 Example 3. Divide by a2, a2c+a2x. The greatest common measure is c+x. cx+x2 Example 4. By dividing one of the quantities by 2, and the other by 3, each becomes, x3-8x-3, and is the great. est common measure. Example 5. First dividing a5-b2a3 by a3, it becomes a a2-b2)a1-b4 (a2+b2 -b2. a4-a2 b2 a2b2-b4 a2 b2-b4 a2-b2 being the last divisor is the greatest common measure. Example 6. First dividing xy+y by y it becomes x+1. x+1)x2-1(x-1 The greatest common measure is x+1. x2+x x—α)x3 — a3 (x2 +ax+a3 Example 8. a2-3ab-2b2)a2-ab-2b2 (1 a2-3ab+2b2 Example 9. a3—a2x—ax2+x3)α1 — x1 (a Dividing this remainder by x, a3x+a2x2 -ax3. Dividing this remainder by 2x, -2a3x+2x3 —a2+x2)a3+a2x—ax2 —x3 (—a Dividing by x, Therefore a2-x2 is the greatest common measure sought. It frequently happens that the common measure of quantities of this kind is better found by resolving both numerator and denominator into their component factors, in which it will be useful to remember that the difference of any two even powers is divisible both by the difference and sum of their roots; and that the difference of two odd powers is divisible by the difference of their roots, and their sum by the sum of their roots; (See Art. 235.) thus in the example just a4x4 (a2+x2)X(a2x2) considered, · (a2—x2)a—(a2 —x2 )x a3. -a2x-ax2+x3 = where it is obvious that a2-x2 is a common measure of both terms, and employing it as a divisor, the fraction may which is in its lowest terms, and con be reduced to α-x sequently a2-2 is the greatest common measure as was found by the first operation. Example 10. First dividing the divisor by a, it becomes a2-b2)a2+2ab+b2 (1 INVOLUTION AND EXPANSION OF BINOMIALS. Art. 475. 3 Example 5. What is the sixth power of (3x+2y)? By substi. a and b, a® +ab+a+b2+a3b3+a3ba +ab3+ the terms. a®+6a5b+15a b2+20a2b3+15a2b1+6α the terms with co-efficients. 36 +35×2×6+3a × 22X15+33×23×20+3 ×24X15+3X25+6+26. the numbers with co-efficients. b+b = Art. 482. Example 2d. Expand into a series (1+x)*. |