Chapter 3.4, Problem 69E

To determine: Why every polynomial equation with real coefficients of degree 3 must have at least one real root.

Solution: A polynomial equation with real coefficients of degree 3 must have at least on real root.

Explanation:

According to Fundamental Theorem of Algebra. $n^{th}$ degree polynomial has exactly $n^{th}$ roots. Also, if $f\left(x\right)$ is a polynomial with real coefficients and $x=a+ib$ solution of $f\left(x\right)=0$ then $x=a-ib$ is also a solution of $f\left(x\right)=0$. Hence, we can say that a cubic equation can have one real root and two complex roots or it can have three real roots Therefore, a cubic equation must have at least one real root.