- The polynomial \(\mathrm { F } ( x )\) is a quartic such that
$$\mathrm { F } ( x ) = p x ^ { 4 } + q x ^ { 3 } + 2 x ^ { 2 } + r x + s$$
where \(p , q , r\) and \(s\) are distinct constants.
Determine the number of possible quartics given that
- the constants \(p , q , r\) and \(s\) belong to the set \(\{ - 4 , - 2,1,3,5 \}\)
- the constants \(p , q , r\) and \(s\) belong to the set \(\{ - 4 , - 2,0,1,3,5 \}\)
(ii) A 3-digit positive integer \(N = a b c\) has the following properties
- \(N\) is divisible by 11
- the sum of the digits of \(N\) is even
- \(N \equiv 8 \bmod 9\)
- Use the first two properties to show that
$$a - b + c = 0$$ - Hence determine all possible integers \(N\), showing all your working and reasoning.