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Chapter 2: Problem 4

If \(a+b i\) is a complex zero of a polynomial with real coefficients, then so is its _____, \(a-b i\).

Short Answer

Expert verified
The correct term to fill in the blank is 'conjugate'. So, the sentence should read: If \(a+b i\) is a complex zero of a polynomial with real coefficients, then so is its conjugate, \(a-b i\).

Step by step solution

01

Understand Complex Numbers

Firstly, it is important to know what complex numbers are. Complex numbers are numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, with the property that \(i^2 = -1\). So, \(a+bi\) and \(a-bi\) are indeed complex numbers.
02

Recognizing the Parantheses

Let's focus on the parentheses in the sentence. The first complex number is \(a+bi\), and the complex number in the blank is \(a-bi\). The change between these two complex numbers is that the sign of \(bi\) has been flipped.
03

Understanding the Theorem

Based on the complex conjugate theorem, if a complex number is a root of a polynomial with real coefficients, then its complex conjugate must also be a root. The complex conjugate of a complex number \(a+bi\) is given by \(a-bi\). So, the blank should be filled with 'conjugate'.

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