$f(x)=x^8-101x^7+101x^6-101x^5+...+101x^2-101x+25$
$=x^8-100x^7-x^7+100x^6+x^6-100x^5-x^5+...+100x^2+x^2-100x-x+25$
$=x^7(x-100)-x^6(x-100)+x^5(x-100)-x^4(x-100)+x^3(x-100)-x^2(x-100)+x(x-100)-x+25$
$=(x-100)(x^7-x^6+x^5-x^4+x^3-x^2+x)-x+25$
$→f(100)=(100-100)(100^7-100^6+100^5-100^4+100^3-100^2+100)-100+25=0-75=-75$