Let $k , n$ and $r$ be positive integers. (a) Let $Q ( x ) = x ^ { k } + a _ { 1 } x ^ { k + 1 } + \cdots + a _ { n } x ^ { k + n }$ be a polynomial with real coefficients. Show that the function $\frac { Q ( x ) } { x ^ { k } }$ is strictly positive for all real $x$ satisfying $$0 < | x | < \frac { 1 } { 1 + \sum _ { i = 1 } ^ { n } \left| a _ { i } \right| }$$ (b) Let $P ( x ) = b _ { 0 } + b _ { 1 } x + \cdots + b _ { r } x ^ { r }$ be a non-zero polynomial with real coefficients. Let $m$ be the smallest number such that $b _ { m } \neq 0$. Prove that the graph of $y = P ( x )$ cuts the $x$-axis at the origin (i.e. $P$ changes sign at $x = 0$) if and only if $m$ is an odd integer.
Let $k , n$ and $r$ be positive integers.
(a) Let $Q ( x ) = x ^ { k } + a _ { 1 } x ^ { k + 1 } + \cdots + a _ { n } x ^ { k + n }$ be a polynomial with real coefficients. Show that the function $\frac { Q ( x ) } { x ^ { k } }$ is strictly positive for all real $x$ satisfying
$$0 < | x | < \frac { 1 } { 1 + \sum _ { i = 1 } ^ { n } \left| a _ { i } \right| }$$
(b) Let $P ( x ) = b _ { 0 } + b _ { 1 } x + \cdots + b _ { r } x ^ { r }$ be a non-zero polynomial with real coefficients. Let $m$ be the smallest number such that $b _ { m } \neq 0$. Prove that the graph of $y = P ( x )$ cuts the $x$-axis at the origin (i.e. $P$ changes sign at $x = 0$) if and only if $m$ is an odd integer.