Determine whether each statement is true or false. If false, explain why. A polynomial function having degree 6 and only real coefficients may have no real zeros.

검증된 단계별 안내

Recall the Fundamental Theorem of Algebra, which states that a polynomial of degree \(n\) has exactly \(n\) roots (zeros) in the complex number system, counting multiplicities.

Since the polynomial has degree 6, it must have exactly 6 roots in total, but these roots can be real or complex.

Because the polynomial has only real coefficients, any non-real complex roots must occur in conjugate pairs. This means complex roots come in pairs like \(a + bi\) and \(a - bi\).

If the polynomial had no real zeros, then all 6 roots would be complex and must come in conjugate pairs. Since 6 is even, it is possible to have 3 pairs of complex conjugate roots and no real roots.

Therefore, it is possible for a degree 6 polynomial with real coefficients to have no real zeros, making the statement true.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Polynomial Degree and Zeros

The degree of a polynomial indicates the highest power of the variable and determines the maximum number of zeros (roots) the polynomial can have. A polynomial of degree 6 can have up to 6 zeros, counting multiplicities and including complex zeros.

Real and Complex Zeros

Polynomials with real coefficients may have zeros that are real or complex. Complex zeros occur in conjugate pairs, meaning if a + bi is a zero, then a - bi is also a zero. This ensures the polynomial's coefficients remain real.

Fundamental Theorem of Algebra

This theorem states that every non-constant polynomial has at least one complex root. For polynomials with real coefficients, all roots (real or complex) sum to the degree of the polynomial, but it is possible for a polynomial to have no real roots if all zeros are complex conjugates.