cmi-entrance 2012 QB1

cmi-entrance · India · ugmath 10 marks Roots of polynomials Divisibility and minimal polynomial arguments
a) Find a polynomial $p ( x )$ with real coefficients such that $p ( \sqrt { 2 } + i ) = 0$. b) Find a polynomial $q ( x )$ with rational coefficients and having the smallest possible degree such that $q ( \sqrt { 2 } + i ) = 0$. Show that any other polynomial with rational coefficients and having $\sqrt { 2 } + i$ as a root has $q ( x )$ as a factor. 
a) Find a polynomial $p ( x )$ with real coefficients such that $p ( \sqrt { 2 } + i ) = 0$.\\
b) Find a polynomial $q ( x )$ with rational coefficients and having the smallest possible degree such that $q ( \sqrt { 2 } + i ) = 0$. Show that any other polynomial with rational coefficients and having $\sqrt { 2 } + i$ as a root has $q ( x )$ as a factor.
Paper Questions
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