RULE II. When one or both of the Factors are compound terms, maltiply the Multiplicand, successively by the term or terms in the multiplier; adding the products together. A NUMERICAL PROOF.-(Invented by the Author.) If the signs of the terms of the Multiplicand are like the signs of the terms in the Multiplier, place the siga + before the product, but if the signs are unlike, place the sign before it. EXAMPLES. 1 + 12 To multiply any quantity by another quantity, is to ADD * the Multiplicand to itself as often as there are units, and parts of an unit in the Multiplier, if the sign of the Multiplier be +. But if the sign of the Multiplier be —, subtract the Multiplicand as often as there are unites and parts of an unit in the Multiplier. To Multiply by Thus, +4* +4 Proof, +4 Here we have +2 in the Multiplier, and we ADD The Multiplicand twice. Here we have -2 in the Multiplier, and we SUBTRACT the Multiplicand trice. Whatever is the number of the Multiplier, the Multiplicand must be ADDED so many times. See Smith's Algebra. Now observe, that in Multiplication, if the MULTIPLICAND be Multiplied by the Multiplier, or the the MULTIPLIER De Multiplied by the MULTIPLICAND, the product will be the same. Therefore, 4 × +2, or +2 x the product will +4 × −2, or −2 × +4) be the same, -3 x x +y. or +yx —x = −xy, and xy = To Multiply -4 by -2 is to Subtract 4 twice, or, to Subtract S, which by the DEMONSTRATION of the Rule of Bubtraction is -S. DIVISION being the converse (contrary) RULE to Multiplication, the RULE is the converse to Multiplication. To the product of the Coefficients we join the sum of the Powers of the letters. RULE I. Divide the coefficient of the Dividend by the coefficient of the divisor; join the difference of the Indices * to the Quotient. The proof that +4 ÷ by −2, or ➡4 + by +2 -4÷by-2 deduced immediately from Multiplication; i. e. that in Division the Product of the Divisor and Quotient = the DIVIDEND, Therefore, +4÷by-2, or 4 by +2 =—2, Because the Divisor 2 x the Quotient 2 +4, the Dividend, Or the Divisor +2 × by the Quotient -24, the Dividend, Because the Quotient +2 x the Divisor➡2 No. 1. 16x2y2 † 96 x3y* 11=3 -4, the Dividend. No. 2. 3 ax | 18 ax2 6t No. 3.-ba2c | 25 a3 c2 No. 4. 6a4y3x2 | 48 a3 y*x* -5 ac 8 ayx RULE II. WHEN THE DIVISOR and DIVIDEND ARE COMPOUND TERMS, Find how often the first term of the Divisor is contained in the first term of thes Dividend. Multiply the whole Divisor by the Quotient, subtract the products from the Dividend, and bring down the next term, and proceed as before. EXAMPLE and PROOF. 3x+2y) 9x2 + 12xy +4y2 (3x + 2y, Answer. 9x2 + 6xy N. B These fourteen pages are only an abridgment of the first four Rules of SMITH'S INTRODUCTION to ALGEBRA. His EASY INTRODUCTION includes QUADRATIC EQUATIONS, and has copious Notes and Elucidations, so that a youth who understands common Arithmetic, may make himself master of this science without any assistance.-Price, in boards, 7s, PRINTED, BY T. SMITH, 105, Fenchurch Street. |