Ex. 5. x2+2x−1(x*. x2 + x − 1 ) x − x * + x2 — x2 + 2 x − 1 ( x* —x3 + x°— x+1 Ex. 7. Divide a*+4a3b+6a2b2+4 ab3+b1 by a+b. ANSWER, a3+3 a3b + 3 a b2 + b3, Ex. 8. . . . . a-5 a1x+10a3x2-10a2x2+5α x2-x3 Ex. 9. by a3-3a2x+3 a x2 + x3. . . . . 25 x-x1-2x3-8x by 5x3-4x3. ANSW. a-2ax➡x2. ANSW. 5x+4x2+3x+2. ANSW. a3+6ax+12ax2+8x3. Ex. 11. Ex. 10. . . . . a* +8 a3x +24 a2x2+32ax2+16x* by a+2x. Ex. 13. .... 9x-46x+95x+150x by x2-4r-5. On the application of the foregoing Rules to Quantities with literal Coefficients. 27. In applying the foregoing rules to quantities with literal coefficients, such as, mx, ny, qx2 &c. (where m, n, q &c. may be considered as the coefficients of x, y, x2, &c.) a compound quantity may be expressed by placing the coefficients of like quantities one after another (with their proper signs) in a parenthesis, and then annexing the common letter or letters. Thus, the sum of mx and nx, which is mx+nx, may be expressed by (m+n)x; their difference, which is mx-nx, by (m—n)x; multinomial m x2+nx2−px2+qx2, by (m+n− p+q)x'; and the mixed multinomial pxy+qy3-rxy+my2nxy, by (p—r—n) x ; +(q+m) y3; &c. &c. According to this method of notation, the operations are performed in the following Examples. (m−p+q)y2 + (n+m)y+(1+n—v−q) z. From Ex. 2. px3+qx2-rx+s Subtract mx3-nx2+tx-v Remainder (p-m) x3 + (q + n) x2 - (r+t)x+s+v.®) Product mpx+(mq−np) x2 - (mr+nq)x+nr. Product ax-(b+ac) x3+(c+ b c + a) x3 − ( c2 + b) x + c. Ex. 5. (Division.) -cx + 1 ) ax• — (b + ac) x3 + (c+bc+a) x2 — (c2+b)x+c(ax® — bx + c аха -acx3 མི bx (a) As the sign prefixed to quantities in a parenthesis affects them all; when this sign is negative, the signs of all those quantities must be changed in putting them into the parenthesis. Thus, when + is sub tracted Ex. 6. Multiply mx2-nx-r ... by nx-r. ANSWER, mnx3 — (n2 + m r) x2 + r2. Ex. 7. Multiply x3-px2+qx−r. by x-a. 3 ANSW. x-(a+p) x2 + (q+ap) x2 −(r-aq)x+ar. Ex. 8. Multiply px-rx+q... by x-rx-q. ANSW. px-(1+p) rx2+(q+r2−pq)x2— q2. Ex. 9. Divide a x3 — (a2 + b) x2+b2. by ax-b. ANSW. x-ax-b. VIII. Some general Theorems, deduced by means of the foregoing Rules. From the clear and distinct manner in which quantity and its several relations are represented throughout every part of an Algebraic operation, the exemplification of its most ordinary rules affords the means of investigating certain general Theorems relating to the sum, difference, product, &c. &c. of numbers, of which the following are examples. 28. Let a and b be any two numbers of which a is the greater and the lesser, and let their sum be represented by s and their difference by d, tracted from -rx, the result is -rx-tx; and, as this means that the sum of rx and tx is to be subtracted, that negative sum is expressed by —(r x + t x ) = −(r+t)x. For the same reason, any multinomial quantity ~m x2+n x2 — qx2+rx2, when put into a parenthesis with a negative sign prefixed, becomes (m-n+q−r)x2. From which we deduce this general Theorem, that " if the sum "and difference of any two numbers be given, the greater of "them may be found by adding half the given sum to half the "given difference; and the lesser, by subtracting half the given "difference from half the given sum." 29. Let a, b, s, d have the same relation as before, then s=a+b d=a-b Hence, by Multiplication, s × d=a2— b2 (See Ex.2. CASE III. p.14.) From which it appears, that "if the sum and difference of any two numbers be multiplied together, the product of that "sum and difference gives the difference of the squares of the 66 two numbers;" and, that "if the difference of the squares " of the two numbers be divided by their difference, it gives their sum; and if by their sum, it gives their difference." 30. Let the number c be divided into any two parts a and b, Then c = a+b c = a + b .. by Multiplication, c2=a2+2ab+b (See Ex. 1. CASE III. p.14.) From which we infer, that " if a number be divided into two parts, the square of the number is equal to the sum of the squares of the two parts, together with twice the product of "those parts." E 31. Let |