# Prove Coefficients of Complex Polynomials are Real

• de1irious
P(z) are real if and only if P(x) is real for all real x.In summary, the coefficients of a complex polynomial P(z) are real if and only if P(x) is real for all real x. This can be shown by comparing the Taylor series expansion of P(x) and P(z*) and exploiting the fact that x and P(x) are real.
de1irious
I need to show that the coefficients of a complex polynomial P(z) are real iff P(x) is real for all real x. Thanks!

Try a Taylor series about 0.

try to look at P(x) and p(x)* where x is real (what do you know about z and z* if they are real). And write

a + b x + c x^2 + ... d x^n

and

(a + b x + c x^2 + ... d x^n)*

and compare them exploiting that x and p(x) is real. If this isn't enough i have made a spoiler in the bottom of the post, just follow the dots.
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a + b x + c x^2 + ... d x^n = real = real* = (a + b x + c x^2 + ... + d x^n)* = a* + b* x* + c* x*^2 + ... + d* x*^n = a* + b* x + c* x^2 + ... + d* x^n

so

0 = (a-a*) + (b-b*)x + (c-c*)x^2 + ... + (d-d*)x^n for all x

so

a=a*, b=b*, c=c* ... d=d*

Last edited:

## 1. What are complex polynomials?

Complex polynomials are algebraic expressions that involve one or more complex variables, such as z. They can be written in the form P(z) = anzn + an-1zn-1 + ... + a1z + a0, where a is a complex number and n is a positive integer. They are used to model and solve problems in various branches of mathematics and science.

## 2. How do you prove that coefficients of complex polynomials are real?

The coefficients of a complex polynomial are real if and only if the polynomial is a real polynomial. To prove this, we can use the fact that a complex number a + bi is real if and only if b = 0. Therefore, for each term in the polynomial, we can set the imaginary part equal to 0 and solve for the coefficient. If all coefficients are real, then the polynomial is a real polynomial and the coefficients are real.

## 3. What is the significance of proving that coefficients of complex polynomials are real?

Proving that coefficients of complex polynomials are real is important because it allows us to simplify and manipulate these polynomials in a more efficient way. It also helps us to understand the properties and behavior of complex polynomials, which have many applications in fields such as physics, engineering, and computer science.

## 4. Can complex polynomials have both real and imaginary coefficients?

Yes, complex polynomials can have both real and imaginary coefficients. In fact, most complex polynomials will have a combination of both, unless they are specifically constructed to have only real coefficients. However, if a complex polynomial has only real coefficients, then it is considered a real polynomial.

## 5. What are some common techniques used to prove that coefficients of complex polynomials are real?

There are several techniques that can be used to prove that coefficients of complex polynomials are real, such as setting the imaginary part equal to 0, using the properties of complex conjugates, and using the Fundamental Theorem of Algebra. Other techniques may involve using mathematical induction or the method of contradiction. The specific technique used will depend on the complexity of the polynomial and the desired level of rigor in the proof.

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