Cubic Polynomial Explorer
Polynomials · Class IX
Shape y = ax^3 + bx^2 + cx + d and observe up-to-three real zeros.
- p(0)0
- p(1)-2
- Derivative roots discriminant36
- y = a x^3 + b x^2 + c x + d
Formulas in this lab
- p(0)
- p(1)
- Derivative roots discriminant
Frequently asked questions
▶What is special about a cubic polynomial?
A cubic polynomial $p(x) = ax^3 + bx^2 + cx + d$ has degree three and can have up to three real zeros. Its graph always crosses the x-axis at least once because of the odd degree. The sum of all three zeros equals $-\frac{b}{a}$.
▶How do I use this lab?
Slide a, b, c, d and observe how the S-shaped curve bends. Try a = 1, b = 0, c = -3, d = 0 to see three zeros at $x = 0, \sqrt{3}, -\sqrt{3}$. Flatten by reducing the leading coefficient.
▶Common mistake on this topic
Some students think every cubic has three distinct real zeros. A cubic can also have one real zero plus two complex conjugate zeros. Always check the graph or use the discriminant idea before claiming three real roots.
▶Where do cubics show up in real life?
The volume of a cube of side x is $V = x^3$, a simple cubic. Engineers fitting cost-versus-output curves for small factories often use cubic models. The lab helps you visualise how shape changes with each coefficient.