# Complex Analysis Proof showing that a Polynomial is linear

• QuantumLuck
In summary, we are given a polynomial P(z) such that P(z) is real if and only if z is real. Our goal is to prove that P(z) is linear. Using the hint provided, we set P = u + iv and z = x+iy, and note that v = 0 if and only if y = 0. From this, we can conclude that 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. Additionally, 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
QuantumLuck

## Homework Statement

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.

## Homework Equations

Cauchy-Reimann equations.
Fundamental Theorem of Algebra (FTA) (i think)

## 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).

anyone have any ideas? after thinking some more i realize that b will follow from cauchy reimann equations so long as i have a. but, i fail to see how a, b, and c together define the polynomial P(z) to be linear or how part a is true.

Hmmm... I'm not sure off the top of my head, but notice P(z) being real when z is real isn't a very enlightening statement, as any real polynomial satisfies this. So the meat of the proof must lie in P(z) being real only when z is real

It's really not clear to me how to prove your hint a). But here's another approach. First show P(z) is a polynomial in z with real coefficients. Now from your knowledge of real polynomials, there is a real number A such that P(x)=A has either one or zero REAL roots (depending on whether the highest power of x is odd or even. That means if it's degree is greater than one, it must have a complex root r. Hence?

Thanks a lot Dick I was able to put together what you said with some of my professor's help to solve the problem. he didn't like the books method either. so i had to show that the power of the polynomial couldn't be even or an odd number that is three or greater. hence, it must be one.

## 1. How do you prove that a polynomial is linear using complex analysis?

To prove that a polynomial is linear using complex analysis, we use the fundamental theorem of algebra which states that every polynomial of degree n has exactly n complex roots. If a polynomial has only one root, then it is a linear polynomial.

## 2. Can a polynomial with complex coefficients be linear?

Yes, a polynomial with complex coefficients can be linear. A linear polynomial is defined as a polynomial of degree 1, and it can have any type of coefficients, whether real or complex.

## 3. What is the role of the complex plane in proving the linearity of a polynomial?

The complex plane plays a crucial role in proving the linearity of a polynomial. We use the complex plane to graph the polynomial and identify the roots of the polynomial. If the polynomial has only one root, then it is linear.

## 4. Can a polynomial of degree higher than 1 be linear?

No, a polynomial of degree higher than 1 cannot be linear. A linear polynomial is defined as a polynomial of degree 1, and a polynomial of degree higher than 1 will have more than one root, making it non-linear.

## 5. Are there any disadvantages to using complex analysis to prove the linearity of a polynomial?

One possible disadvantage of using complex analysis to prove the linearity of a polynomial is that it requires a strong understanding of complex numbers and their properties. It may also involve more complex mathematical calculations compared to other methods of proving linearity.

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