try to solve but failed..

## 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

Tutors, please sign in to answer this question.

# 2 Answers

Let P(a) be a polynomial in a, such that

7a

^{6}-2a^{5}-3a^{4}+6a^{3}-4a^{2}+6a-P(a)=3a^{5}+6a^{4}-5a^{3}+2a^{2}+3Then

7a

^{6}-2a^{5}-3a^{4}+6a^{3}-4a^{2}+6a - (3a^{5}+6a^{4}-5a^{3}+2a^{2}+3) = P(a)So

P(a)=7a

^{6}-5a^{5}-9a^{4}+11a^{3}-6a^{2}+6a-3Let'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.