(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose P is a polynomial such that P(z) is real iff. z is real. Prove that P is linear.

The hint given in the text is to set P = u + iv, z = x+iy and note that v = 0 iff y = 0.

We are then told to conclude that

a. either v-sub y(partial of v with respect to y) is greater than or equal to 0 throughout the real axis or is less than or equal to 0 throughout the real axis;

b. either u-sub x(partial of u with respect to x) is greater than or equal to zero or is less than or equal to zero for all real values and hence u is monotonic along the real-axis;

c. P(z) = alpha has only one solution for real-valued alpha.

2. Relevant equations

Cauchy-Reimann equations.

Fundamental Theorem of Algebra (FTA) (i think)

3. The attempt at a solution

My friend was of the opinion that we can say v > 0 for y > 0, then we must have v < 0 for y < 0 because the FTA says v greater than or equal to 0 for all z is impossible. (yet i do not see how this follows at all and i am not sure the path to a solution that this will bring.

I am not quite sure how the proof is supposed to follow from the given guidelines and any help would be much appreciated (even a hint).

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Complex Analysis Proof showing that a Polynomial is linear

**Physics Forums | Science Articles, Homework Help, Discussion**