Help with this problem

Question

If x⁹+ 512 = (x+2)(a₈x⁸+ a₇x⁷+...+a₁x+a₀) then a₈ + a₇ + ... + a₁ + a₀ = ??

Answer 1

(x⁹+512)=(x+2)(x⁸-2x⁷+4x⁶-8x⁵+16x⁴-32x³+64x²-128x+256)

So

1-2+4-8+16-32+64-128+256≠0

a8 + a7 + ... + a1 + a0 ≠0

 

 

Answer 2

x⁹ + 512 = x9 + 2⁹

 

If you use long division of polynomials, you can write

 

x + 2 | x⁹ + 0x⁸ + 0x⁷ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 2⁹

 

where | is my nomenclature for a long division sign. When I do this division I get a very cool answer:

 

x⁸ - 2x⁷ + 2²x⁶ - 2³x⁵ + 2⁴x⁶ - 2⁵x³ + 2⁶x² - 2⁷x + 2⁸