186 



THE POPULAR EDUCATOR. 



LESSONS IN ALGEBRA. VI. 



MULTIPLICATION (continued) . 



RULE FOE SIGNS IN THE PRODUCT. 



1. THE rule is that + into + produces + ; into + gives 

 ; + into gives ; and into - gives + ; or, in words, 

 plus multiplied by plus gives plus ; minus by plus gives minus 

 plus by minus gives minus; and minus by minus givea pins; 

 that is, if the signs of the factors are ALIKE, tlie sign of the pro- 

 duct will be plus, or affirmative; 'but if the signs of the factors are 

 UNLIKE, the sign of the product will be minus, or negative. 



82. The first case, viz., that of + into +, needs no expla- 

 nation, being the same as that of ordinary numbers. 



83. The second case is into +, that is, the multiplicand is 

 negative, and the multiplier positive. Thus, a- into + 4 is 



4a. For the repetitions in the multiplicand are a a a 



a = 4a. 



EXAMPLES. - 



-(1). Multiply 2a m 

 By 37i + x 



(2.) h 3d + 4 



Product : 6ah 3hm + 2ax mx 2hy - 6dy -f 8y 



(3.) Multiply 

 By 



a 2 7d x 

 36 + h 



Product : Sab 66 2lbcl 3bx + ah 2h 7dh - hx. 



84. In the two preceding cases, the positive sign prefixed to 

 the multiplier shows that the repetitions of the multiplicand 

 are to bo added- to the other quantities with which the multi- 

 plier is connected. But in the two remaining cases, the negative 

 .sign prefixed to the multiplier indicates that the sum of the 

 repetitions of the multiplicand are to be subtracted- from the 

 other quantities. This subtraction is performed at the time of 

 multiplying, by making the sign of the product opposite to that 

 of the multiplicand. Thus + a into 4 is 4a. For the 

 .repetitions of the multiplicand are, -\-a-\-a-\-a-\-a -f- 4a. 

 But this sum is to be subtracted from the other quantities with 

 -which the multiplier is connected. It will then become 4ft 

 [Art. 58]. Thus in the expression b (4 X a) it is manifest 

 that 4 X a is to be subtracted from 6. Now 4 X a is 4a, that 

 -.is, -j- 4a. But to subtract this from b, the sign + must be 

 changed into . So that b (4 X a) is b 4a. And a X 4 

 is therefore 4a.. 



Again, suppose the multiplicand is a, and the multiplier 

 16 4). As (6 4) is equal to 2, the product will be equal to 

 J2a. This is less than the product of 6 into a. To obtain, then, 

 the product of the compound multiplier (G 4) into a, wo must 

 subtract the product of the negative part from that of the 

 .positive part. Thus, multiplying a by 6 4 is the same as 

 multiplying a, by 2. And the product of the former, viz., 6a 

 4a, is the same as the product of the latter, viz., 2a. But if 

 the multiplier be (6 + 4), the two products must be added. 

 Thus, multiplying a by G + 4, is the same as multiplying a 

 by 10. And the product of the former, viz., 6a-\-4a, is the 

 same as the product of the latter, viz., 10a. 



This shows at once the difference between multiplying by a 

 positive factor, and multiplying by a negative one. In the 

 .former case, the sum of the repetitions of the multiplicand is 

 to be added to, in the latter it is to be subtracted from, the 

 .other quantities with which the multiplier is connected. 



.EXAMPLES. (1.) Multiply a -f b 

 By b x 



Product : ab -\- b z ax bx. 



(2.) Multiply 



By 



3dy + lix -\- 2 

 mr ab 



Product i 3dmry -\- hmrx + 2mr Sabdy abhx 2ab. 



(3.) Multiply 3h + 3 



By ad 6 



Product .- 3adh -f Sad 18h 18. 



85. If two negatives be multiplied together, the product will 

 be affirmative : 4 X a = + 4a. In this case, as in the 

 preceding, the repetitions of the multiplicand are to be subtracted, 

 because the multiplier lias the negative sign. These repe- 

 .iations, if the multiplicand is a, and the multiplier 4, are 



a a a a = 4a. But this is to be subtracted by 

 changing the sign. It then becomes + 4a. 



Suppose a is multiplied by (6 4). As 6 4=2, the 

 product is evidently twice the multiplicand, that is, 2a. But 

 if we multiply a into 6 and 4 separately, a into 6 is 



6, and a into 4 is 4a [Art. 83]. As in the multi- 

 plier, 4 is to be subtracted from 6 ; so, in the product, 4a 

 must be subtracted from 6a. Now, 4a becomes by sub- 

 traction -f- 4a. The whole product then is 6a + 4a, which is 

 equal to 2a. Or thus, multiplying a by 6 4, is the 

 same as multiplying a by 2 ; and the product of the former, 

 viz., 6a + 4a, is equal to the product of the latter, viz., 2a. 

 Hence the general rule may be thus stated: When quant:!/:':; 

 are multiplied by a positive term, their signs are retained in the 

 product ; but ivhen by a negative one, they are changed. 



86. It is often considered a great mystery that the product 

 of two negatives should be affirmative. But it amounts to 

 nothing more than this, that the subtraetion of a negative 

 quantity is equivalent to the addition of an affirmative one 

 [Arts. 58, 59], and therefore that the repeated subtraction of a 

 negative quantity is equivalent to the repeated addition of an 

 affirmative one. So, taking off from a man's hands a debt of 

 ten pounds every month, is adding ten pounds a month to the 

 value of his property. 



EXAMPLES/ (1.) Multiply a 4 into Bb 6. Answer. 3ab 



I2b Ga + 24. 



(2.) Multiply Sad ah 7 into 4 dy Jir. Answer. 1 2ad 



4ah 28 3ad z y + adhy -f- 7dy 3adhr -f- ali,-r -f- 7hr. 

 (3.) Multiply 2hy + 3m 1 into 4tl 2x + 3. Answer. 8dhy 



+ 12dm 4cJ 4-lix-y Gmx + 2x -f 6hy + Qut, 3. 



87. Positive and negative terms may frequently balance each 

 other, so as to disappear in the product. [Art. 53.] 



(1.) Multiply 

 By 



EXAMPLES. 



(2.) mm yy 



mm -)- yy 



bb +mn 



Product : aa * bb. mmmm 

 (3.) Multiply aa -f ab + bb 



By a b 



aaa -j- Gab -f- abb 



aab abb- bub 



yyyy 

 yyyy- 



aaa * * bbb. 



88. For many purposes it is sufficient merely to indicate the 

 multiplication of compound quantities, without actually multi- 

 plying the several terms. Thus [Art. 23], the product of 



a + b c into h + m -f- y, is (a + b c) X (h -j- m -f- y). 



EXAMPLES. (1.) What is the product of a -f m into h + x 

 and d + y ? Answer, (a + m) (h -j- x) (d -f- y). 



By this method of representing multiplication, a,n important 

 advantage is often gained, in preserving the factors distinct 

 from each other. When the several terms are multiplied in 

 form, the expression is said to be expanded. 



(2.) What does (a + b) x (c + d) become when expanded 9 

 Answer, ac + ad + be + bd. 



89. With a given multiplicand, the less the multiplier, the 

 less will be the product. If, then, the multiplier be reduced 

 to nothing, the product will be nothing. Thus a X = 0. And 

 if be one of any number of fellow-factors, the product of the 

 whole will be nothing. 



EXAMPLES. (1.) What is the product of ab X c X 3d X 0? 

 Answer. 0. 



(2.) And (a + b) X (c + c?) X (h m) X ? Answer. 0. 

 (3.) Multiply 1 -f x -f x" + x 3 -\- x* f v- by 1 x + a; 2 . Ans. 



I + K 2 + W 3 + X* + K 5 + .T 7 . 



(4.) Multiply 1 + x + z~ + x 3 + x* -f ;<;* by 1 x -f- a; 2 x 3 

 + a 4 x 5 . Ans. 1 + x- -f a- 4 a; 6 a- 8 a 110 . 



(5.) Multiply a + 2b + c by a c. An*, a 2 + 2ab 2bc c-. 



(G.) Find the continual product of xy 1, xz 1, and yz 1. 

 Ans. x^ipz" x^yz xy z z w/s 2 -f- xy + a?z + yz 1. 



(7.) Find the continual product of x' 2 -f- yz, y- + xz, and z 2 + xy. 

 Ans. 2x z y' 2 z* + x 3 y 3 -{- x 3 z 3 -f- y 3 z 3 + xyz* + xyx* + x*yz. 



