Prove polynomial divisibility property

3

1. Factorise fully \(n ^ { 3 } - n\).
2. Hence prove that, if \(n\) is an integer, \(n ^ { 3 } - n\) is divisible by 6 .

6

1. Multiply out \(\left( 3 ^ { n } + 1 \right) \left( 3 ^ { n } - 1 \right)\).
2. Hence prove that if \(n\) is a positive integer then \(3 ^ { 2 n } - 1\) is divisible by 8 .

1. (i) A student states
   "if \(x ^ { 2 }\) is greater than 9 then \(x\) must be greater than 3 "

Determine whether or not this statement is true, giving a reason for your answer.
(ii) Prove that for all positive integers \(n\), $$n ^ { 3 } + 3 n ^ { 2 } + 2 n$$ is divisible by 6

3 Prove by mathematical induction that, for all integers \(n \geqslant 1 , n ^ { 5 } - n\) is divisible by 5 .

10
- \(\begin{array} { c } \text { Prove by induction that } \mathrm { f } ( n ) = n ^ { 3 } + 3 n ^ { 2 } + 8 n \text { is divisible by } 6 \text { for all integers } n \geq 1 \\ \text { [7 marks] } \\ \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \end{array}\) -
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