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  • Show that p - 1 is a factor of p^{10} - 1 and also of p^{11} - 1 .

Show that p - 1 is a factor of p10 - 1 and also of p11 - 1 .

Answers (1)

To prove : Here we have to prove that p-1  is a factor of p^{10}-1  and also of p^{11}-1 .

We know that if (x+a)  is factor of the polynomial f(x), then it always satisfies f(-a)=0

If p-1  is a factor of p^{11}-1  and p^{10}-1  then by putting the value of p = 1, the given polynomials should be equal to zero.

Let       

\\f(p)=p^{10}-1\\ f(1)=(1)^{10}-1\\ =1-1=0\\ f(1)=0

Hence p-1  is a factor of p^{10}-1

Let   

\\g(p)=p^{11}-1\\ g(1)=(1)^{11}-1\\ =1-1=0\\ g(1)=0

Hence p-1  is a factor of p^{11}-1

Hence proved

 

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