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what should be subtracted from 7a^6-2a^5-3a^4+6a^3-4a^2+6a to get 3a^5+6a^4-5a^3+2a^2+3

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2 Answers

Let P(a) be a polynomial in a, such that
 
7a6-2a5-3a4+6a3-4a2+6a-P(a)=3a5+6a4-5a3+2a2+3
 
Then
7a6-2a5-3a4+6a3-4a2+6a - (3a5+6a4-5a3+2a2+3) = P(a)
 
So
P(a)=7a6-5a5-9a4+11a3-6a2+6a-3
Let's Assume that the solution is P(a)
 
 
then:
 
   7 a^6 - 2a^5 - 3a^4 + 6 a^3 - 4 a^2 + 6a - p(a) = 3a^5 + 6 a^4 - 5 a^3 + 2 a^2 + 3
 
  p( a) = 7a^6 - 2a^5 - 3a^4 + 6 a^3 - 4 a^2 + 6a - 3a^5 + 6 a^4+5 a^3 -2a^2 -3 =
 
           =   7 a^6 -5 a^5 -9 a^4 + 11a^3- 6 a^2 +6a -3
 
  It is  always helpful to proceed along algebraic framework , choose a variable( function, i.e. P(a) ) for the
 
  unknown  and work with algebra.