Graph and Roots of a Third Degree Polynomial

A third degree equation

ax³ + bx² + cx + d = 0,

with the leading coefficient a ≠ 0, has three roots one of which is always real, the other two are either real or complex, being conjugate in the latter case. In the former case, the two real roots may coincide. It may also happen that all three roots are real and equal.

(In the applet, clicking at the numbers that indicate the window of the graph will rescale the graph. To change the numbers up or down, click a little off their vertical center lines.)

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The reason that a third degree polynomial always has a real root is that its limits as x → ±∞ have different signs (one is +∞, the other -∞) implying that there are always two points at one of which the polynomial is positive while at the other negative. Then the existence of a real root stems from Bolzano's Theorem.

The graph of a third degree polynomial always has an inflection point about which it is symmetric. Subtracting from the polynomial the linear function that described the tangent to its graph at the point of inflection leaves a polynomial with three equal (real) roots. (To see that click in a vicinity of the inflection point. A second click removes the add-ons.)

A curious property has been observed by Audrey Weeks and reported by Lin McMullin.