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\left(p^{4}-7\right)\left(p^{4}+3\right)
Find one factor of the form p^{k}+m, where p^{k} divides the monomial with the highest power p^{8} and m divides the constant factor -21. One such factor is p^{4}-7. Factor the polynomial by dividing it by this factor. The following polynomials are not factored since they do not have any rational roots: p^{4}-7,p^{4}+3.