The cubic keeps three relations

A quadratic has at most 2 zeros. A cubic, with highest power x³, has at most 3.

The rule

A cubic polynomial has the form ax³ + bx² + cx + d, with a not 0.

Suppose the cubic has three zeros α, β and γ. Then:

α + β + γ = −b/a. αβ + βγ + γα = c/a. αβγ = −d/a.

The middle line adds products of two zeros at a time.

One cubic, checked

2x³ − 5x² − x + 6 has zeros 2, −1 and 3/2. First check each one.

Here a = 2, b = −5, c = −1 and d = 6.

The pairs are 2×(−1) = −2, (−1)×(3/2) = −3/2 and (3/2)×2 = 3.

Now try it. Three zeros are shown. Tap the cubic that has them.

Try the lab to finish this lesson.