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Chapter 5: Problem 4

Show that every polynomial of odd degree with real coefficients has at least one real root.

Short Answer

Expert verified
By virtue of the Intermediate Value Theorem and the end behavior of polynomials with odd degree, it can be confirmed that every polynomial of odd degree with real coefficients has at least one real root.

Step by step solution

01

Understand the Properties of Odd Degree Polynomials

An important thing to remember is that the end behaviors of polynomials of odd degree follow a pattern: as \( x \) tends to -鈭�, the polynomial tends to 鈭� or -鈭�; as \( x \) tends to 鈭�, the polynomial will behave inversely compared to its behavior at -鈭�. That is, if it tends to 鈭� at -鈭�, it will tend to -鈭� at 鈭�, and vice versa.
02

Apply Intermediate Value Theorem

The Intermediate Value Theorem states that if a function \( f \) is continuous on a closed interval \[a, b\], and \( n \) is any number between \( f(a) \) and \( f(b) \), then there exists a number \( c \) in the interval \[a, b\] such that \( f(c) = n \). We know that polynomials are continuous everywhere, so all real values of \( x \) are in our interval.
03

Relate the Theorem to the Odd Degree Polynomial

By the properties of polynomials of odd degree, there are numbers a and b such that \( f(a) < 0 \) and \( f(b) > 0 \) (or vice versa). So 0 is a number that lies between \( f(a) \) and \( f(b) \). By the Intermediate Value Theorem, there must be at least one number \( c \) such that \( f(c) = 0 \), meaning the polynomial has at least one real root.

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Most popular questions from this chapter

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