Elements of Algebra

80 ELEMENTS SECT. II. each term of it is separately multiplied by that number; so that the product is $2a - 2b + 2c$. And the like takes place with regard to all other numbers; for if $d$ were the number by which it was required to multiply the same expression, we should obtain $ad - bd + cd$. 272. In the last article, we have supposed $d$ to be a positive number; but if the multiplier were a negative number, as $-e$, the rule formerly given must be applied; namely, that unlike signs multiplied together produce $-$, and like signs $+$. Thus we should have $-ae + be - ce$. 273. Now, in order to shew how a quantity, $A$, is to be multiplied by a compound quantity, $d - e$; let us first consider an example in numbers, supposing that $A$ is to be multiplied by $7 - 3$. Here it is evident, that we are required to take the quadruple of $A$: for if we first take $A$ seven times, it will then be necessary to subtract $3A$ from that product. In general, therefore, if it be required to multiply $A$ by $d - e$, we multiply the quantity $A$ first by $d$, and then by $e$, and subtract this last product from the first: whence results $dA - eA$. If we now suppose $A = a - b$, and that this is the quantity to be multiplied by $d - e$; we shall have $dA = ad - bd$ $eA = ae - be$ whence $dA - eA = ad - bd - ae + be$ is the product required. 274. Since therefore we know accurately the product $(a - b) \times (d - e)$, we shall now exhibit the same example of multiplication under the following form: $a - b$ $d - e$ -------------------- $ad - bd - ae + be$. Which shews, that we must multiply each term of the upper expression by each term of the lower, and that, with regard to the signs, we must strictly observe the rule before given; a rule which this circumstance would completely confirm, if it admitted of the least doubt. 275. It will be easy, therefore, according to this method, to calculate the following example, which is, to multiply $a + b$ by $a - b$;