Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

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.

Step-by-Step Solution

Answer

The statement is true. A third-degree polynomial function with integer coefficients can have exactly one real root and a pair of complex conjugate roots.

Step 1: Understanding the polynomial function

A third-degree polynomial function means it can be written in the form \( ax^3 + bx^2 + cx + d = 0 \), in which \( a, b, c, d \) are integers and \( a \neq 0 \). The number of roots this polynomial can have is exactly 3 according to fundamental theorem of algebra.

Step 2: Understanding the nature of polynomial zeros

The roots of a polynomial function can be either real or complex, and complex roots always appear in conjugate pairs. The roots a+bi and a-bi are considered to be a conjugate pair. Since the coefficients are all integers and we know that the equation must has 3 roots, the roots could be all real or one real and other two a complex conjugate pair.

Step 3: Conclusion

It is indeed possible for a third-degree polynomial function with integer coefficients to have no real zeros, except the case where this polynomial has one real root and a pair of complex conjugate roots, it can't have no real roots at all. So the statement is true.