 Now in this question it says show that x minus 1 is a factor of x to the power minus 1 and also of x to the power 11 minus 1. So this is usually discussed as a theorem but let's say a theorem we are going to discuss in subsequent videos whenever you see such kind of expression how to find out whether something is factor or not but the application has been given early so let's try with whatever knowledge of factor theorem we have to solve this problem. So show that x minus 1 is a factor of x to the power 10 minus 1. So let us say fx is equal to x to the power 10 minus 1 and if x minus 1 is a factor of fx then clearly what will happen? f of 1 must be 0 that is by factor theorem right the value of the function or the polynomial in this case at x equals to 1 must be 0 so let's check whether it is actually 0 so f of 1 is 1 to the power 10 minus 1 and 0 correct it's 0. So hence therefore we can say x minus 1 is a factor of fx that is x to the power 10 minus 1 the other way of writing this as x minus 1 divides x to the power 10 minus 1 okay now let's check for the other one x to the power 11 minus 1 let's say gx is equal to x to the power 11 minus 1 right so what will be g of 1 is again 0 you can check 0 point is 0 therefore x minus 1 divides x to the power 11 minus 1 as well now you can generalize this if you see is it so x minus 1 is a factor of x to the power n minus 1 where where n is a positive integer okay positive integer so whenever n is positive integer that is natural number 1 2 3 4 5 6 whatever then x minus 1 will always be a factor of this y because every time let's say if hx is x to the power n minus 1 therefore h of 1 if you see will always be 0 1 to the power n minus 1 is 0 therefore x minus 1 is a factor of x to the power n minus 1 whatever value whatever positive integer n could be correct so this is the generalized statement