Problem 69 - College Algebra

Question

Why must every polynomial equation with real coefficients of degree 3 have at least one real root?

Step-by-Step Solution

Answer

Every polynomial equation of degree 3 with real coefficients must have at least one real root. This is due to the Fundamental Theorem of Algebra and the fact that complex roots in polynomials with real coefficients always appear in conjugate pairs. The remaining third root of the polynomial has to be real.

Step 1: Analyzing the root possibilities

From the Fundamental Theorem of Algebra, a polynomial of degree 3 has exactly 3 roots within the complex number system. If these roots are all real, we are done. If not, they must be a combination of real and complex numbers.

Step 2: Unpacking the complex roots

Complex roots in polynomials with real coefficients always appear in conjugate pairs, meaning if \(a + bi\) is a root, then \(a - bi\) is also a root. For a cubic polynomial, having two complex roots leaves space for one root that has to be real.

Step 3: Conclusion

Given that complex roots must come in pairs and a polynomial of degree 3 has exactly three roots, it is guaranteed that there should be at least one real root. If there were no real roots, the minimum we could have would be two complex ones which leaves us with only two roots for a third degree polynomial. This contradicts the Fundamental Theorem of Algebra.