The mechanical stress S S\,S measured in a reinforced composite beam depends on a reinforcement parameter k k\,k and is modeled by the function:
S(k)=10k3+9k2+10k+4 S(k) = 10k^3 + 9k^2 + 10k + 4 S(k)=10k3+9k2+10k+4Use the factor theorem to show that (2k+1)(2k + 1)(2k+1) is a factor of S(k)S(k)S(k).
Express S(k)S(k)S(k) in the form
S(k)=(2k+1)(ak2+bk+c) S(k) = (2k + 1)(ak^2 + bk + c) S(k)=(2k+1)(ak2+bk+c)where aaa, b b\,b and c c\,c are integers to be determined.
Given that n n\,n is a positive integer, use your result from part (b) to determine whether 10n3+9n2+10n+410n^3 + 9n^2 + 10n + 410n3+9n2+10n+4 can ever be a prime number. Justify your answer.
