1.3.1 Operations on polynomials

A discrete model for the total number of stable configurations, $N (k)$ , of a crystal lattice with $k\,$ layers is given by

N(k) = 10k^3 + 11k^2 + 7k + 2

Use the factor theorem to show that $(2 k + 1)$ is a factor of $N (k)$ .

[2]

Express $N (k)$ in the form

N(k) = (2k + 1)(ak^2 + bk + c)

where $a$ , $b\,$ and $c\,$ are constants to be found.

[3]

Given that $n\,$ is a positive integer, use your answer to part (b) to explain why $10n^3 + 11n^2 + 7n + 2$ is never prime.

[2]

1.3.1 Operations on polynomials Questions

A discrete model for the total number of stable configurations, $N (k)$ , of a crystal lattice with $k\,$ layers is given by

N(k) = 10k^3 + 11k^2 + 7k + 2

Use the factor theorem to show that $(2 k + 1)$ is a factor of $N (k)$ .

[2]

Express $N (k)$ in the form

N(k) = (2k + 1)(ak^2 + bk + c)

where $a$ , $b\,$ and $c\,$ are constants to be found.

[3]

Given that $n\,$ is a positive integer, use your answer to part (b) to explain why $10n^3 + 11n^2 + 7n + 2$ is never prime.

[2]