Given that f(x)=(x-2)^5+2(x-2)^4+3(x-2)^3+4(x-2)^2+5(x-2)+6f(x)=(x−2)5+2(x−2)4+3(x−2)3+4(x−2)2+5(x−2)+6. If f(x)=a(color(red)(x+2))^5+b(x-2)^4+c(x-2)^3+d(x-2)^2+e(x-2)+ff(x)=a(x+2)5+b(x−2)4+c(x−2)3+d(x−2)2+e(x−2)+f. Find the values of a,b,c,d,e,fa,b,c,d,e,f ?

Note that the question is a(x+2)^5a(x+2)5 and not a(x-2)^5a(x−2)5 , which makes me have no clue how to solve it...

1 Answer
Cesareo R.
Jun 26, 2017

See below.

Explanation:

Calling

f(x)=(x-2)^5+2(x-2)^4+3(x-2)^3+4(x-2)^2+5(x-2)+6f(x)=(x−2)5+2(x−2)4+3(x−2)3+4(x−2)2+5(x−2)+6
g(x)=a_0(x+2)^5+b_0(x-2)^4+c_0(x-2)^3+d_0(x-2)^2+e_0(x-2)+f_0g(x)=a0(x+2)5+b0(x−2)4+c0(x−2)3+d0(x−2)2+e0(x−2)+f0
we have

{(f(2)=g(2)->6-a_0 4^5-f_0), (d/(dx)f(2)=d/(dx)g(2)->5 - 5*4^4 a_0 - e_0 = 0), (d^2/(dx^2)f(2)=d^2/(dx^2)g(2)->8 - 1280 a_0 - 2 d_0 = 0), (d^3/(dx^3)f(2)=d^3/(dx^3)g(2)->18 - 960 a_0 - 6 c_0 = 0), (d^4/(dx^4)f(2)=d^4/(dx^4)g(2)-> 48 - 480 a_0 - 24 b_0 = 0), (d^5/(dx^5)f(2)=d^5/(dx^5)g(2)->120 - 120 a_0 = 0):}

Solving for a_0,b_0,c_0,d_0,e_0,f_0 we obtain

a_0=1,b_0=-18,c_0=-157,d_0=-636,e_0=-1275,f_0=-1018

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