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

Question

try to solve but failed..

Answer 1

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

Answer 2

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.