Rewrite a polynomial function in a power of (x-2).

Question

I've been looking at my notes completely stumped trying to figure out how to approach this question, the question is as follows:

Write $f(x) = 3x^5 - 2x^4 + x^3 - 5x^2 - 7x + 11$ in the power of (x-2).

Answer 1

Hint:

use Taylor series at $x=2$ $$f(x)=f(2)+{\frac {f'(2)}{1!}}(x-2)+{\frac {f''(2)}{2!}}(x-2)^{2}+{\frac {f'''(2)}{3!}}(x-2)^{3}+\cdots .$$

Answer 2

Let $y=x-2$. Then we have a polynomial in $x=y+2$, $f(x)=3(y+2)^5−2(y+2)^4+(y+2)^3−5(y+2)^2−7(y+2)+11$. Now expand everything out. (Note, this may end up being more work than the hint of @E.H.E.)

Answer 3

Hint: write $f(x+2) = 3(x+2)^5 - 2(x+2)^4 + (x+2)^3 - 5(x+2)^2 - 7(x+2)x + 11$. Expand the powers and collect the terms to get $f(x+2)=3 x^5 + a x^4 + b x^3 + c x^2 + d x + e$. Then $f(x)=3 (x-2)^5 + a (x-2)^4 + b (x-2)^3 + c (x-2)^2 + d (x-2) + e$.