Sliding the Roots of Cubics

coefficient of

2

x

0

coefficient of x

2.

constant term

-1

The roots of a cubic polynomial depend on the coefficients of the cubic in a complicated way. In this Demonstration, you move the roots in the complex plane by varying the coefficients of the cubic.

If the coefficients

b

,

c

, and

d

of a cubic

3

x

+b

2

x

+cx+d

are real, the cubic will have either three real roots or one real root and a pair of roots that are complex conjugates of each other. For some combinations of coefficients, two roots will slide along the real axis, then merge (forming a double root), then split and move off the real axis to become a pair of complex conjugate roots.