Algebra Question #10

$So\quad yes\quad this\quad is\quad not\quad a\quad Number\quad Theory\quad question.\\ \\ We\quad let\quad x\quad =\quad { 2 }^{ 100 }\\ \\ Our\quad divisor\quad =\quad g(x)\quad =\quad x\quad -\quad 1\\ \\ Now\quad the\quad polynomial\quad to\quad divide:-\\ \\ p(x)\quad =\quad { 8 }^{ 101 }\quad +\quad { 4 }^{ 101 }\quad +\quad { 2 }^{ 101 }\quad +\quad 1\\ \quad \quad \quad \quad =\quad 8.{ 8 }^{ 100 }\quad +\quad 4.{ 4 }^{ 100 }\quad +\quad 2.{ 2 }^{ 100 }\quad +\quad 1\\ \quad \quad \quad \quad =\quad 8{ x }^{ 3 }\quad +\quad 4{ x }^{ 2 }\quad +\quad 2x\quad +\quad 1\\ \\ Using\quad Remainder\quad theorem,\quad we\quad put\quad x\quad =\quad 1\\ \\ \therefore \quad p(1)\quad =\quad 8.{ 1 }^{ 3 }\quad +\quad 4.{ 1 }^{ 2 }\quad +\quad 2.1\quad +\quad 1\\ \quad \quad \quad \quad \quad \quad \quad =\quad 8\quad +\quad 4\quad +\quad 2\quad +\quad 1\\ \quad \quad \quad \quad \quad \quad \quad =\quad \boxed { 15 } \\$

(8^{101} + 4^{101} + 2^{101} + 1) / 2^{100} - 1 =( 8(8^100) + 4(4^100) + 2(2^100) + 1)/2^{100} - 1. Let x = 2^100, so ----> (8x^3 + 4x^2 + 2x + 1)/ x - 1. By Remainder Theorem ---> 8(1)^3 + 4(1)^2 + 2(1) +1 = 15 answer

a=2^100, A=a-1, B=8a^3+4a^2+2a+1; B/A do the division, reminder would be 15.