Learning Resources

Shapes of Polynomial Functions

• Remember: Imaginary roots occur in pairs! Thus, two is the maximum number of imaginary roots that a cubic can have. As a result. every cubic must have at least one real root. It may indeed have three real roots, but it is guaranteed to have at least one. It stands to reason that this would be true for all polynomials of odd degree. 
• As for the quartic, there is no guarantee that any of its roots are real. What we do know is that there are indeed four roots. They may all be real, all imaginary or two may be real and the other two imaginary. Since imaginary roots occur in pairs, it would not make sense to conclude that one root is real and the other three are imaginary. There must always be an even number of imaginary roots existing, if indeed any exist at all.