More Nested Stuff
5 + 7 + 5 + 7 + 5 + … 6 6
Let f ( x ) denote the least degree monic polynomial with integer coefficients with root equal to the number above. Find the value of f ( 1 ) .
The answer is -38.
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1 solution
You could circumvent the second (less elegant) solution as follows: Require that the value of the expression is a root of a monic polynomial with rational coefficients.
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True. With the condition that the degree is minimal among those that meet the requirements. Otherwise ( x 3 + 7 x 2 − 5 x − 4 1 ) n would suffice for any natural n .
You have a very Dutch name for someone living in the USA by the way; any Dutch roots? :)
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Yes, Dutch born :) I lived in Amsterdam as a student, from 1995 to 2003.
If x is a value of the expression, we have:
x 2 − 5 = 7 + x 6
i.e.
x 3 + 7 x 2 − 5 x − 4 1 = 0 .
Therefore a 0 + a 1 + a 2 + a 3 = 1 + 7 − 5 − 4 1 = − 3 8 .
However : the roots of this polynomial are approximately equal to − 6 . 8 5 7 , − 2 . 5 1 8 and 2 . 3 7 5 . The former two don't qualify, seeing as x is a square root and therefore x ≥ 0 . Therefore the minimum polynomial is actually y = x − 2 . 3 7 5 , meaning a 0 + a 1 = − 1 . 3 7 5 .