1. If $P$ and $Q$ are polynomial functions with no common factors, then $y = Q(x)/P(x)$ is analytic at $x_0$ if and only if $P(x_0) \not= 0$. Moreover the radius of convergence of the power series for $ Q(x)/P(x) $ about the point $x_0$ is

$min\{||x_0 - x|| ~|~ x \in {\bf C}, P(x) = 0\}$

where $||x_0-x||$ = distance from $x_0$ to $x$ in the complex plane.

               A) True                        

2. The power series for $f(x) = -(2 + x)$ converges for all $x$.

               A) True
Note the power series for $f(x) = -2 - x + 0x^2 + 0x^3 + 0x^4 + ... $                        

3.) $ \left[\matrix{5 \cr 6}\right], \left[\matrix{4 \cr 13}\right]$ are linearly independent.

               A) True                        

4.) $ \left[\matrix{5 \cr 6}\right], \left[\matrix{4 \cr 13}\right], \left[\matrix{1 \cr 1}\right]$ are linearly independent.

                                        B) False

5.) $ \left[\matrix{5 \cr 6}\right], \left[\matrix{-10 \cr -12}\right]$ are linearly independent.

                                        B) False

Answers