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Chapter 2: Problem 115

Decide whether the statement is true or false. Justify your answer. It is possible for a third-degree polynomial function with integer coefficients to have no real zeros.

Short Answer

Expert verified
The statement is False. A third-degree polynomial function with integer coefficients always has at least one real zero.

Step by step solution

01

Recognize The Statement

The exercise is asking whether it's possible for a third-degree polynomial function with integer coefficients to have no real zeros. This is a question about the Fundamental Theorem of Algebra.
02

Understanding The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-constant polynomial equation has at least one complex root, which includes real roots as a subset. Hence, a polynomial of degree \(n\) will have exactly \(n\) roots, though these roots may be complex (a+b*i, where both 'a' and 'b' are real numbers), and may not be distinct.
03

Compare and Review The Statement

In the context of the given problem, a third-degree polynomial will have exactly three roots (not necessarily distinct or real). However, it cannot be true that a third-degree polynomial has no real zeros. Even if two of the roots are complex (non-real), the third root will always be real. This is due to the Complex Conjugate Root Theorem, which states that non-real roots of polynomials with real coefficients always occur in conjugate pairs.
04

Conclusion

Hence, given the mathematical principles, one can conclude that the statement provided in the exercise is False. It is not possible for a third-degree polynomial function with integer coefficients to have no real zeros. At least one real root is ensured by the Fundamental Theorem of Algebra and the Complex Conjugate Root Theorem.

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Most popular questions from this chapter

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=16 x^{3}-20 x^{2}-4 x+15$$

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