Chapter 3 - Polynomial and Rational Functions - Section 3.2 The Real Zeros of a Polynomial Function - 3.2 Assess Your Understanding - Page 225: 109

$-7$

The Remainder theorem states that the remainder of a function $f(x)$ after dividing by $(x-r)$ is $f(r)$. As per the given equation, when $f(x)=2x^{20}-8x^{10}+x-2$ is divided by $x-1$, then by the remainder theorem, $f(1)$ represents the remainder of the function. We simplify the given equation as follows: $f(1)=2(1)^{20}-8(1)^{10}+1-2\\=2-8+1-2 \\=-7$