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Chapter 2: Problem 89

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Every polynomial equation of degree 3 with real coefficients has at least one real root.

Short Answer

Expert verified
The given statement is True.

Step by step solution

01

Understand what each term means

A Real Coefficient is a real number that is multiplied by a variable. The Degree of a Polynomial is the highest power of a variable in the polynomial. A Root of a Polynomial is a solution to the equation when set equal to zero.
02

Check the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every polynomial of degree n has exactly n roots. Some of these roots may be real and some may be complex.
03

Verify the statement

Every polynomial equation of degree 3 with real coefficients has roots that may be real or complex. However, there is a special case for polynomials of odd degree. They always cross the x-axis at least once and therefore, always have at least one real root.
04

Final verification

So, the given statement that 'Every polynomial equation of degree 3 with real coefficients has at least one real root' is indeed True.

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