- #106

SWest1234

- 1

- 1

We can factor the polynomial 5x^3+6x^2+5x+4 as(x+1)(5x^2+x+4).

5x^2+x+4 >x+1 for all x. so let x+1=p^a and 5x^2+x+4=p^b where b>a. Divided, we obtain (5x^2+x+4)/(x+1)=p^b-a, which is an integer. This can be written as 5x-4+8/(x+1). For this to be an integer, we must have x+1 dividing 8, which means the only solution is base 7, in which case we get 56547=204810 which equals 2^11.