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How can i solve this quickly and intuitively? I can certainly bang out the roots but i want to know the shortcut
BELOW IS REVISED VERSION OF THIS QUESTION:
If \(t\) is a prime number, is \(32t^3 - 16t^2 + 8t - 4\) divisible by \(t^2\)?
Since first two terms of \(32t^3 - 16t^2 + 8t - 4\) are divisible by \(t^2\) then the question becomes whether \(8t-4\) is divisible by \(t^2\).
(1) \(t^2 <25\) --> since \(t\) is a prime number then \(t=2\) or \(t=3\). If \(t=2\) then \(8t-4\) is divisible by \(t^2=4\) but if \(t=3\) then \(8t-4\) is NOT divisible by \(t^2=9\). Not sufficient.
(2) \(t^2-8t+12=0\) --> \(t=2\) (\(t=6\) is not a valid solution since \(6\) is not a prime number). Sufficient.