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Chapter 3: Problem 69

Why must every polynomial equation with real coefficients of degree 3 have at least one real root?

Short Answer

Expert verified
Because of the Fundamental Theorem of Algebra, a polynomial of degree 3 must have 3 roots, which could be real or complex. If a polynomial has real coefficients, all non-real roots are complex numbers and come in conjugate pairs. Since the degree is an odd number, there must be at least one real root.

Step by step solution

01

Understanding the Fundamental Theorem of Algebra

First, it is important to know the Fundamental Theorem of Algebra which states that every non-constant polynomial equation of degree n has n roots in the complex number plane. These roots can be real or complex.
02

Real Coefficients and Their Roots

Second, it's critical to understand that if a polynomial has real coefficients and a complex root, the conjugate of that complex root is also a root. So, complex roots for polynomials with real coefficients show up in conjugate pairs.
03

Analyzing Degree of Polynomial

Our polynomial is a cubic polynomial, which means its degree is 3 (an odd number).
04

Concluding the Analysis

Since the cubic polynomial has 3 roots and complex roots show up in pairs, it cannot have 3 complex roots. Therefore, a cubic polynomial must have at least one real root.

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