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.
Examples of Cubic Curves And Equations
Example 1 Graph of y = f(x) = (x+8)(x+10)(x+20) = x^{3} + 38x^{2} + 440x + 1600 A few computed points on the curve, apart from the zero(s) which are known: (21,143), (18,160), (15,175), (12,64), (9,11), (6,112), (3,595)
If the given cubic function is: f(x) = ax^{3 }+ bx^{2 }+ cx + d The derivative of this function is: f'(x) = 3ax^{2} + 2bx + cThe function given to us us f(x) = (x+8)(x+10)(x+20) = x^{3} + 38x^{2} + 440x + 1600 And the derivative for this is f'(x) = 3x^{2} + 76x + 1
Consider the cubic equation f(x) = (x+8)(x+10)(x+20) = x^{3} + 38x^{2} + 440x + 1600 = 0 The roots of this cubic equation are at: (x  (8)) = 0 => x = 8, OR (x  (10)) = 0 => x = 10, OR (x  (20)) = 0 => x =20 This cubic equation has real and unique roots at 8, 10, 20. The plotted curve cuts the xaxis at these values of x: i.e, these are the zeroes of the given cubic polynomial
The turning or stationary points is where f'(x) = 0 => 3x^{2} + 76x + 1 = 0 => x = 16.38, x = 8.95 These are also called the "critical" points where the derivative is zero. Coming to other geometrical features of this curve: The yintercept of this curve is at y=1600. And the second derivative of this curve becomes zero at x = 12.67. At this point the curve changes concavity. A cubic curve has point symmetry around the point of inflection or inflexion.
The zeroes of a polynomial, if they are known, and the coefficients of that polynomial are two different sets of numbers that have interesting relations. 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.
Example 2
Graph of y = f(x) = (x+14)(x 0)(x+20) = x^{3} + 34x^{2} + 280x + 0 A few computed points on the curve, apart from the zero(s) which are known: (21,147), (18,144), (15,75), (12,192), (9,495), (6,672), (3,561), (0,0), (3,1173)
If the given cubic function is: f(x) = ax^{3 }+ bx^{2 }+ cx + d The derivative of this function is: f'(x) = 3ax^{2} + 2bx + cThe function given to us us f(x) = (x+14)(x 0)(x+20) = x^{3} + 34x^{2} + 280x + 0 And the derivative for this is f'(x) = 3x^{2} + 68x + 1
Consider the cubic equation f(x) = (x+14)(x 0)(x+20) = x^{3} + 34x^{2} + 280x + 0 = 0 The roots of this cubic equation are at: (x  (14)) = 0 => x = 14, OR (x  (0)) = 0 => x = 0, OR (x  (20)) = 0 => x =20 This cubic equation has real and unique roots at 14, 0, 20. The plotted curve cuts the xaxis at these values of x: i.e, these are the zeroes of the given cubic polynomial
The turning or stationary points is where f'(x) = 0 => 3x^{2} + 68x + 1 = 0 => x = 17.26, x = 5.41 These are also called the "critical" points where the derivative is zero. Coming to other geometrical features of this curve: The yintercept of this curve is at y=0. And the second derivative of this curve becomes zero at x = 11.33. At this point the curve changes concavity. A cubic curve has point symmetry around the point of inflection or inflexion.
The zeroes of a polynomial, if they are known, and the coefficients of that polynomial are two different sets of numbers that have interesting relations. 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
Example 3
Graph of y = f(x) = (x 4)(x+10)(x+20) = x^{3} + 26x^{2} + 80x 800 A few computed points on the curve, apart from the zero(s) which are known: (21,275), (18,352), (15,475), (12,256), (9,143), (6,560), (3,833), (0,800), (3,299), (6,832), (9,2755)
If the given cubic function is: f(x) = ax^{3 }+ bx^{2 }+ cx + d The derivative of this function is: f'(x) = 3ax^{2} + 2bx + cThe function given to us us f(x) = (x 4)(x+10)(x+20) = x^{3} + 26x^{2} + 80x 800 And the derivative for this is f'(x) = 3x^{2} + 52x + 1
Consider the cubic equation f(x) = (x 4)(x+10)(x+20) = x^{3} + 26x^{2} + 80x 800 = 0 The roots of this cubic equation are at: (x  (4)) = 0 => x = 4, OR (x  (10)) = 0 => x = 10, OR (x  (20)) = 0 => x =20 This cubic equation has real and unique roots at 4, 10, 20. The plotted curve cuts the xaxis at these values of x: i.e, these are the zeroes of the given cubic polynomial
The turning or stationary points is where f'(x) = 0 => 3x^{2} + 52x + 1 = 0 => x = 15.63, x = 1.71 These are also called the "critical" points where the derivative is zero. Coming to other geometrical features of this curve: The yintercept of this curve is at y=800. And the second derivative of this curve becomes zero at x = 8.67. At this point the curve changes concavity.
Example 4
Graph of y = f(x) = (x 10)(x+20)(x+20) = x^{3} + 30x^{2} + 0x 4000 A few computed points on the curve, apart from the zero(s) which are known: (21,31), (18,112), (15,625), (12,1408), (9,2299), (6,3136), (3,3757), (0,4000), (3,3703), (6,2704), (9,841), (12,2048), (15,6125)
If the given cubic function is: f(x) = ax^{3 }+ bx^{2 }+ cx + d The derivative of this function is: f'(x) = 3ax^{2} + 2bx + cThe function given to us us f(x) = (x 10)(x+20)(x+20) = x^{3} + 30x^{2} + 0x 4000 And the derivative for this is f'(x) = 3x^{2} + 60x + 1
Consider the cubic equation f(x) = (x 10)(x+20)(x+20) = x^{3} + 30x^{2} + 0x 4000 = 0 The roots of this cubic equation are at: (x  (10)) = 0 => x = 10, OR (x  (20)) = 0 => x = 20, OR (x  (20)) = 0 => x =20 This cubic equation has a double root at x = 20. So the plotted cubic curve has not only a zero for that value of x, but the first derivative at that point is also zero. A tangent to the curve drawn at that point will be parallel to the xaxis.
The turning or stationary points is where f'(x) = 0 => 3x^{2} + 60x + 1 = 0 => x = 20.0, x = 0.0 These are also called the "critical" points where the derivative is zero. Coming to other geometrical features of this curve: The yintercept of this curve is at y=4000. And the second derivative of this curve becomes zero at x = 10.0. At this point the curve changes concavity. A cubic curve has point symmetry around the point of inflection or inflexion.
The zeroes of a polynomial, if they are known, and the coefficients of that polynomial are two different sets of numbers that have interesting relations. 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
Example 5
Graph of y = f(x) = (x+20)(x+20)(x+20) = x^{3} + 60x^{2} + 1200x + 8000 A few computed points on the curve, apart from the zero(s) which are known: (21,1), (18,8), (15,125)
If the given cubic function is: f(x) = ax^{3 }+ bx^{2 }+ cx + d The derivative of this function is: f'(x) = 3ax^{2} + 2bx + cThe function given to us us f(x) = (x+20)(x+20)(x+20) = x^{3} + 60x^{2} + 1200x + 8000 And the derivative for this is f'(x) = 3x^{2} + 120x + 1
Consider the cubic equation f(x) = (x+20)(x+20)(x+20) = x^{3} + 60x^{2} + 1200x + 8000 = 0 The roots of this cubic equation are at: (x  (20)) = 0 => x = 20, OR (x  (20)) = 0 => x = 20, OR (x  (20)) = 0 => x =20 This cubic equation has a triple root at x = 20. So the plotted cubic curve has not only a zero for that value of x, but the derivative and second derivative at the point are also zero.
The turning or stationary points is where f'(x) = 0 => 3x^{2} + 120x + 1 = 0 => x = 20.0, x = 20.0 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. And the second derivative of this curve becomes zero at x = 20.0. At this point the curve changes concavity. A cubic curve has point symmetry around the point of inflection or inflexion. These are just some of the important features and aspects to keep in mind while trying to visualize and analyze a plot of an algebraic function. A graphical and visual inspection helps in several ways.
The zeroes of a polynomial, if they are known, and the coefficients of that polynomial are two different sets of numbers that have interesting relations. 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
Here is a large number of cubic curves which you may study and observe. Curve sketching is an important requirement in many high school exams of Asia, UK, US  and is common, in ISC, IB, CBSE question papers, as well as entrance examinations like the IIT JEE. Curve sketching is a handy tool, used both directly and indrectrly in these examinations.
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Curve/Cubic Equation for zeros, roots (2,0,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (2,0,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (2,0,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (2,0,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (2,0,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (2,0,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (2,10,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (2,10,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (2,10,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (2,10,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (2,10,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (2,10,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (2,10,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,20,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,20,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,20,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,20,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,20,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,20,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,20,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,10,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,10,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,10,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,10,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,10,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,10,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,10,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,0,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,0,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,0,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,0,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,0,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,0,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,0,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,10,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,10,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,10,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,10,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,10,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,10,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (1,10,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,20,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,20,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,20,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,20,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,20,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,20,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,20,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,10,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,10,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,10,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,10,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,10,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,10,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,10,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,0,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,0,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,0,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,0,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,0,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,0,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,0,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,10,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,10,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,10,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,10,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,10,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,10,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (4,10,16)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, 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 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 (7,0,20)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,0,14)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,0,8)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,0,2)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,0,4)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,0,10)Graph plot of Cubic Polynomial Function Curve/Cubic Equation for zeros, roots (7,0,16)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 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) 
