Example  I.    Find  the  cube  root  of

By  inspection,  the  cube  root  begins  with  x'  and  ends  with  —1.
Assume  that  a^+px^+qx^l  is  the  cube  root.
«(x»+ia>)»+...  +  (««-l)»

Equating  the  coefficients  of  a^  and  of  x,  we  see  that  the  only  possible
exact  cube  root  is  a^~x*— x~l.
This  should  now  be  cubed  to  make  sure  that  the  original  expression
is  an  exact  cube.
There  is  a  method  similar  to  Method  III  for  square  roots.
In  this  the  trial  divisor  is  three  times  the  squara  of  the  portion
of  the  root  already  fpund,  to  this  we  add  three  times  the  product
of  the  portion  already  found  and  the  new  term,  together  with  the
square  of  the  new  term.
Example  II.    Find  the  cube  root  of

o«  +  6a»  +  15a*  +  20a»  +  15a«  +  6a  +  l.
o" + 2  a  + 1    cube  root.
a«+6a"  +  15a*  +  20a»  +  15a«  +  6a+l

8a*+  6a»+  4a«  6a'^+16a*  +  20o«

3aH12a»  +  12a»  6a'^-fl2a«+  8a»

3a*  +  12a'+15a'  +  6a  +  l  8a*  +  12a''  +  15a«  +  6a  +  l

Cube  root    a* + 2  a  + 1.
Examples  20011  o.
1.  Find  the  fectors  of  6a?*-7ay-8y»-8a?+y+2.
2.  Show  that  the   graph   of  6«*  +  7a;y-8y»-4a:+5y-2- 0
consiBts  of  two  straight  lines.    Draw  the  graph.
3.  Find  the  value  of  c  in  order  that  2a?»  +  iry-6y'  +  8a?-4y+c
may  be  resolved  into  factors..
b2


848  INDETERMINATE  COEFFICIENTS

4.  Find  the  values  of  h  and  c  in  order  that  there  shall  be  no
remainder  when  «*+6ir'— 20«'  +  &r+<j  is  divided  by  a:*  — 3a; +  2.
5.  Find  the  H.  C.  F.  and  L.  C.  M.  of

«*-2«'+2«-l  and  2iB'-3«*  +  l.
