A discrete model for the total number of stable configurations, N(k)N(k)N(k), of a crystal lattice with k k\,k layers is given by
N(k)=10k3+11k2+7k+2 N(k) = 10k^3 + 11k^2 + 7k + 2 N(k)=10k3+11k2+7k+2Use the factor theorem to show that (2k+1)(2k + 1)(2k+1) is a factor of N(k)N(k)N(k).
Express N(k)N(k)N(k) in the form
N(k)=(2k+1)(ak2+bk+c) N(k) = (2k + 1)(ak^2 + bk + c) N(k)=(2k+1)(ak2+bk+c)where aaa, b b\,b and c c\,c are constants to be found.
Given that n n\,n is a positive integer, use your answer to part (b) to explain why 10n3+11n2+7n+210n^3 + 11n^2 + 7n + 210n3+11n2+7n+2 is never prime.
Practise Edexcel A Level Maths Algebraic Methods with exam-style questions for A Level Maths. 226 questions covering 1.1 Proof by Contradiction, 1.2 Algebraic Fractions, 1.3 Partial Fractions, 1.4 Repeated Factors, and 1.5 Algebraic Division, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
