Question 5
Let
$$
f(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \cdots + a_{0} \quad \text{and} \quad g(x) = c_{n+1} x^{n+1} + c_{n} x^{n} + \cdots + c_{0}
$$
be non-zero polynomials with real coefficients such that $g(x) = (x + r) f(x)$ for some real number $r$. If $a = \max \left(\left|a_{n}\right|, \ldots, \left|a_{0}\right|\right)$ and $c = \max \left(\left|c_{n+1}\right|, \ldots, \left|c_{0}\right|\right)$, prove that $\frac{a}{c} \leq n+1$.
$$
f(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \cdots + a_{0} \quad \text{and} \quad g(x) = c_{n+1} x^{n+1} + c_{n} x^{n} + \cdots + c_{0}
$$
be non-zero polynomials with real coefficients such that $g(x) = (x + r) f(x)$ for some real number $r$. If $a = \max \left(\left|a_{n}\right|, \ldots, \left|a_{0}\right|\right)$ and $c = \max \left(\left|c_{n+1}\right|, \ldots, \left|c_{0}\right|\right)$, prove that $\frac{a}{c} \leq n+1$.