ALGEBRA 



191 



ALGEBRA 



It should be borne in mind that in these 

 problems in subtraction the algebraic sum of 

 each subtrahend and remainder equals its min- 

 uend. These are therefore the correct solutions 

 of the problems given above. 



It is advisable to learn the shortest method 

 to use in subtraction and to know the simplest 

 rule to be applied. By examining the above 

 problems we see that in each case we could 

 have found the same remainder if we had 

 imagined the signs in the subtrahends to have 

 been changed and the minuends and subtra- 

 h.-u.ls then added. Apply the following rule to 

 of the three solutions: 



ige the problem so that like terms in the 

 minuend and subtrahend will be one above the 

 other; change all the signs in the subtrahend 

 from + to and from to + and proceed 

 as in addition. The result will be the remainder 

 sought. 



Multiplication. We have already learned 

 that when we write down any number of alge- 

 braic quantities together without joining them 

 by the plus and minus signs we indicate mul- 

 tiplication. That is, a times b=ab. When we 

 set down graphically the product of abed and 

 bc*dy we find that b is taken twice as a factor, 

 c three times, a once, y once and d twice. The 

 result of our multiplication, in expanded form, 

 is abbcccddy, or, simplified, ab*c*dPy. 



The small figures at the right of and slightly 

 above the letters are known as exponents ; each 

 indicates the number of times the letter is to 

 be used as a factor; 6 2 means the square of b, 

 that is, 6 multiplied by itself or raised to the 

 second power. When a letter is written with- 

 out any exponent, as b, we understand that 

 tin- first power of the letter is meant. That is, 

 6=6*. It is clear. thru, that when like (plan- 

 ties are multiplied, their exponents are ad<l< <1. 

 Thus, 6X6=6*+ 1 =6 S . But o*X6*=a6 s . We 

 can combine exponents of like quantities only. 



Now let us take a more complicated problem 

 for solution: 



56*c +2d 



Let us analyze another problem. Multiply 

 5a-b by 3a. Multiplying these quantities is 

 equivalent to subtracting 5a 2 6 3a times. It 

 must be remembered, however, that in sub- 

 traction the sign of the subtrahend is alway> 

 changed; so, in subtracting 5a 2 6 3a times, \u 

 have the equivalent of adding 5a 2 6 3a times, or 

 of adding the product of 5a 2 b and 3a once. 

 Therefore the product is ISa 3 ^. 



Note the results in the following, where the 

 operations are placed side by side: 



5a 2 6 

 3a 



15a 3 6 



5a 2 6 

 3a 



5a 2 6 5a 2 6 

 3a 3a 



-15a 3 6 15a 3 6 15a 3 6 



It is evident, from the above, that when tin- 

 signs in the multiplier and multiplicand are 

 alike the product is a positive quantity and has 

 the sign +; when the signs in the multiplier 

 and multiplicand are unlike the product is a 

 negative quantity and has the sign . 



The following solution indicates the steps 

 taken when the multiplier and multiplicand 

 have more than one term: 



a 2 2a6+6' 

 a b 



a 3 2a 2 6+a6 2 



The following problems may be solved for 

 practice : 



1. Multiply 4o 36 by 3a+46. 



2. Multiply a 2 a6+6 by 3a+ 6. 



3. Multiply * 2 +2z+ y by x y. 



4. Multiply x<+2z 2 y 2 +i/* by x 1 y*. 

 Division. We learned that in multiplying, 



exponents of like terms in the multiplier and 

 multiplicand are added; in division, which is 

 the reverse of multiplication, the quotient is 

 obtained by subtracting the exponents of lik 

 terms in dividend and divisor. 



Divide 6 3 by 6. 



It happens that the signs in this problem are 



all +. Let us see what steps to take when 



minus signs OTCMI-. I-'nul the product of 60*6 



and 3a. Since 50*6 indicates that 5a*6 is to 



be subtracted, then multiplying 5a*6 by 3a 



. same as subtracting 5a*6 3a times, or sub- 



tract inn th. product of 5a*6 and 3a once. The 



' 15a*6. 



Proof: 6X6=6 3 . Also, 6'H-6=6*-i=6*. 

 Tin- iii\ i .-ion may also be shown thus: 

 Divide 6' by 6. 

 6=666. 



Divide 666 by 6. 

 666 



