Ex. 11. Multiply 3a+2ab-b... by 3a2-2ab+b. ANSW. 9a-4a2b2+4 ab3-b1. Ex. 12. . . . . . x2+x2y+xy2+y3 by x-y. 23. In the Division of algebraic quantities, the same circumstances are to be taken into consideration as in their multiplication, and consequently the four following Rules must be observed. 1. That if the signs of the dividend and divisor be like, then the sign of the quotient will be +; if unlike, then the sign of the quotient will be -.* II. That the coefficient of the dividend is to be divided by the coefficient of the divisor, to obtain the coefficient of the quotient. III. That all the letters common to both the dividend and the divisor must be rejected in the quotient. IV. That The Rule for the signs follows immediately from that in Multiplication; thus, IV. That if the same letter be found in both the dividend and divisor with different indices, then the index of that letter in the divisor must be subtracted from its index in the dividend, to obtain its index in the quotient. Thus, +abc ... or =+b. +ac 6abc or =-3bc. -2a -10xyz =-2x2. +5y Of Division, also, there are three Cases; the same as in Multiplication. CASE I. 24. When dividend and divisor are both simple terms. 25. When the dividend is a compound quantity, and the divisor a simple one; then each term of the dividend must be divided separately, and the resulting quantities will be the quotient required. Divide 90 ax3-18ax2+4a3x-2ax by 2ax. 90 ax3-18a x2+4a2x-2 ax 2ax =45ax-9x+2a-1. Ex. 3. Divide 4x3- 2x2 + 2x by 2x. 4x3-2x2+2x 2x Ex. 4. Divide 24 a'xy-3axy+6x'y' by — 3xy. -3xy Ex. 5. Divide 14 ab3+7 a2b2 — 21 al2+35a3b by 7ab. 14a3+7ab2-21 ab3 +35 ab 7ab CASE III. 26. When dividend and divisor are both compound quantities. In this case, the Rule is, "to arrange both dividend and "divisor according to the powers of the same letter, beginning "with the highest; then find how often the first term of the "divisor is contained in the first term of the dividend, and 66 place the result in the quotient; multiply each term of the "divisor by this quantity, and place the product under the "corresponding (i. e. like) terms in the dividend, and then "subtract it from them; to the remainder bring down as many terms of the dividend, as will make its number of terms "equal "equal to that of the divisor; and then proceed as before, till "all the terms of the dividend are brought down, as in common "arithmetic." In this Example, the dividend is arranged according to the powers of a, the first term of the divisor. we proceed by the following steps; Having done this, 1. a is contained in a3, a' times; put this in the quotient. 11. Multiply a-b by a', and it gives a3-a'b. III. Subtract a3-a'b from a3-3ab, and the remainder is-2a'b. iv. Bring down the next term +3ab'. v. a is contained in -2ab, -2ab times; put this in the quotient. VI. Multiply and subtract as before, and the remainder is a b2. VII. Bring down the last term - b3. VIII. a is contained in ab', +6 times; put this in the quotient. IX. Multiply and subtract as before, and nothing remains ; the quotient therefore is a-2ab+b2. a2+2a Ex. 2. a2+2ax+x2) a3+5a3x+10a3x2+10a3x3+5ax1+x3 ( a3+3a2x+3ax2+x3 4x2 -7x) 12x3 — 13x*—34x2+39x2 (3x2+2x2 — 5x+ 4x3-7x) 4x2 (a) 4x-7x (a) When there is a remainder, it must be made the numerator of a Fraction whose denominator is the divisor; this Fraction must then be placed in the quotient (with its proper sign), the same as in common Arithmetic. |