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Chapter 2: Problem 121

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. According to the Fundamental Theorem of Algebra, a third-degree polynomial function with integer coefficients must have at least one real zero.

Step by step solution

01

Understand the statement

The problem asks if a third-degree polynomial function with integer coefficients can have no real zeros. Here, the degree of polynomial (3) and the specific type of coefficients (integer) are crucial.
02

Apply the Fundamental Theorem of Algebra

According to the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n complex roots. Also, a nonzero polynomial degree with real coefficients has n roots, real or complex.
03

Compare the information from the theorem to the statement

In the statement, the polynomial function is of degree 3, thus it will have exactly 3 complex roots.
04

Final reasoning

The roots can be real or complex but there must be at least one real root because complex roots occur in pairs. Hence, the statement is false.

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