cmi-entrance 2023 QB3

cmi-entrance · India · ugmath 13 marks Roots of polynomials Location and bounds on roots

Suppose that for a given polynomial $p ( x ) = x ^ { 4 } + a x ^ { 3 } + b x ^ { 2 } + c x + d$, there is exactly one real number $r$ such that $p ( r ) = 0$.
(a) If $a, b, c, d$ are rational, show that $r$ must be rational.
(b) If $a, b, c, d$ are integers, show that $r$ must be an integer.
Possible hint: Also consider the roots of the derivative $p ^ { \prime } ( x )$.

Suppose that for a given polynomial $p ( x ) = x ^ { 4 } + a x ^ { 3 } + b x ^ { 2 } + c x + d$, there is exactly one real number $r$ such that $p ( r ) = 0$.\\
(a) If $a, b, c, d$ are rational, show that $r$ must be rational.\\
(b) If $a, b, c, d$ are integers, show that $r$ must be an integer.

Possible hint: Also consider the roots of the derivative $p ^ { \prime } ( x )$.

Paper Questions

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