SOLUTIOSS TO THE EXERCISES.] M ATHEM A TICS. ALGEBR A. 495 



1L Cancelling x, from dividend and divisor, and removing the vincula, the operation is as follows : 

 3-r 3)6 3 + 9x= 20(2*' 



15x 2 20 

 IS* 2 15* 



15j- 20 

 15x 15 



4 Rem. 

 12. *- 



r ax 



a*-^-pa-{-g Rem. 



a 3 -4- pa 3 -4- 0a -4- r 

 13. x_ a) x'+^'+ ?J ; + r(*'+(a+p) J : + a 5 + f a + ? + - 



0)4: a' /?o* ja 



a* +/>a" + jo + r Rem. 



The final remainder in Example 12 is a 1 + pa + <7. and that in Example 13 is a 3 +pa* + ?a + r ; in each case 

 the remainder is the very same expression as we should get by rubbing out the x in the dividend and putting a 

 instead, as observed at page 462. The property is perfectly general : if any polynomial proceeding according to 

 the powers of x be divided by z a, the remainder will always bo the same as the polynomial itself when the x is 

 replaced by a. Hence, when a is any given number, you will be able to ascertain, without actual division, whether 

 a proposed polynomial is divisible by z a or not: thus, suppose it were asked Is x* 2-e 35 divisible by 

 z 7 ? Putting 7 for x in x 1 2x 35, wo get 49 14 35 = 0. Hence, the remainder is nothing, so that the 

 expression is divisible by z 1. (See Ex. 1). 



In like manner, referring to Ex. 2, and substituting 3 for x in x"- x 12, it becomes 9 + 3 12 = 0. Hence, 

 z* x 12 is divisible by z +3. And a glance will serve to show that the division may be performed without 

 remainder in Ex. 7. 



SQUARE ROOT o A SIMPLE QUANTITY. EXAMPLES FOE EXERCISE. PAGE 466. 



1. -/aVy = aVv'y. 2. \J 2V4' = -f 2aA'. 3. 



4. \J 8a'.V = 2orW- 5 - ^ ~ 2V* = V* 4 ^ 2. 6. ^Ztfy't = a'y' 



7. ('4V)* = **'*. I.^AVss.fVk 9. 



10. Vo 1 ^ - = ax- v'a. 11. 



12 / ?0 ' - /"2^__J?. /^-H_^?L. 13. (-8 



V 27TV - V O.S^'y* - 3^y' V 3* - 3*y 3* 



J_ 1 1 ^y 



14. (9&'.*V) -7M\?p-34Vy^-3Vp' 



, _ - _ _ 



la. **(8I*V ) - ^/81*V* ~ 3T'y 3 i 'V 3 *** ~3y 3 *" V 3 *** 



16. \J - ' - J ' 



-J - J _i ar> a^'V 16 a**\/2 



18.(32.-'*-V) '=32 'ax'y =-=--=-3-. 



KEDUCTION OF FRACTIOXS. EXAMPLES FOR KXERCISE. PAGE 467. 



a 1 2 y^ _ 9 y' 2 + y __ 7_ 



- 2. 3-y- 3 >y- 3+y ~ 3 -j- y 



3 ' U V * (A c) "(A c) r 



a 5 -f x* 2x 3 



* -^.-^ =, by actual division, a' + ax + x' + a _ f - 



* By actual division, ~jrn 4TT == B * 



. 



2*y 2^y 2xy 



