Let us consider the cubic function f(x) = (x+20)(x+20)(x+20) = x^{3}^{2} We will inspect the graph, the zeroes, the turning and inflection points in the cubic curve curve y = f(x).
Cubic Polynomials and Equations
A cubic polynomial is a polynomial of degree 3.
where a is nonzero. An equation involving a cubic polynomial is called a cubic equation and is of the form f(x) = 0. There is also a closedform solution known as the cubic formula which exists for the solutions of an arbitrary cubic equation. A cubic polynomial is represented by a function of the form. And f(x) = 0 is a cubic equation. The points at which this curve cuts the Xaxis are the roots of the equation.
Graph of y = f(x) = (x+20)(x+20)(x+20) = x^{3}^{2}
The function given to us us f(x) = (x+20)(x+20)(x+20) = x^{3}^{2}
Consider the cubic equation f(x) = (x+20)(x+20)(x+20) = x^{3}^{2} The roots of this cubic equation are at: (x  (20)) = 0 => x = 20, OR OR This cubic equation has a triple root at x = 20.
The turning or stationary points is where f'(x) = 0 => 3x^{2} These are also called the "critical" points where the derivative is zero. Coming to other geometrical features of this curve: What we see here is the graph of a nonlinear function. The yintercept of this curve is at y=8000.
If we know the zeroes, then we can write down algebraic expressions for the coefficients. Going the other way is much harder and cannot be done in general. A cubic function has a bit more variety in its shape than the quadratic polynomials which are always parabolas. We can get a lot of information from the factorization of a cubic function. We get a fairly generic cubic shape when we have three distinct linear factors
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roots (7,20,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,20,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,20,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,20,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,20,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,20,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,20,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,10,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,10,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,10,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,10,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,10,4)Graph plot of Cubic Polynomial 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(7,10,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,10,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,10,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,10,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,10,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,20,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,20,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,20,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,20,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,20,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,20,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,20,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,10,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,10,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,10,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,10,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,10,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,10,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,10,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,0,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,0,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,0,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,0,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,0,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,0,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,0,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,10,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,10,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,10,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,10,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,10,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,10,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (10,10,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,20,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,20,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,20,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,20,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,20,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,20,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,20,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,10,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,10,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,10,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,10,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,10,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,10,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,10,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,0,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,0,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,0,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,0,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,0,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,0,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,0,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,10,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,10,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,10,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,10,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,10,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,10,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (13,10,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,20,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,20,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,20,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,20,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,20,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,20,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,20,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,10,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,10,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,10,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,10,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,10,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,10,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,10,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,0,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,0,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,0,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,0,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,0,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,0,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,0,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,10,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,10,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,10,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,10,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,10,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,10,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (16,10,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,20,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,20,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,20,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,20,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,20,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,20,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,20,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,10,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,10,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,10,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,10,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,10,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,10,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,10,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,0,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,0,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,0,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,0,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,0,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,0,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,0,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,10,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,10,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,10,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,10,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,10,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,10,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (19,10,16) 
