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Cubic Equations


The polynomials can be of the following types: linear polynomial, quadratic polynomial and the cubic Polynomials. If we talk about the Cubic Equations, we say that the standard form of the cubic Equation is ax3 + b.x2 + cx + d = 0, where we have a, b, c and d as the real Numbers and a <> 0.
Now we look at the Cubic Equations and the method of solving the Cubic Equations. The Cubic Equations can be solved by finding the common factor of the given equation and then dividing the given equation by the common factor. In case the resultant factor i.e. the quotient is the quadratic polynomial, then we say that the solution of this can be done by any of the methods of solving the quadratic equations. Here we will observe the relation between the zeros and the coefficients of cubic equations.

Let α, β, γ are the three zeroes of the given cubic polynomial. We know that the cubic equation can have maximum of three zeroes. So if we have p(x) = ax3 + b.x2 + cx + d, where a < > 0, then we say that (x-α ) ,( x - β) ,( x – γ) are the factors of p(x).
So we have
ax3 + b.x2 + cx + d = k. (x - α) * (x - β) * (x – γ) for some constant ‘k’.
On comparing we get k = a, -k * (α + β + γ) = b, k * (α* β + β* γ + γ * α) = c and -k (α* β* γ) = d.
Putting the value of k = a, we get (α + β + γ) = b/ a,
(α* β + β* γ + γ * α) = c/a and
α* β* γ = - d/ a.

Solving Cubic Equation

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In Algebra, we come across several types of equations like linear equation, polynomial equations, quadratic equation and many more. In short we can say that in algebra revolves around different equations and its solutions. Whenever we have to solve any problem in algebra we first try to construct its equation and then solve that equation. We all must be aware of cubes and Cube roots, when we talk about cube it means thrice of that number or three time multiplication of that number. So a cube equation is an equation that involves cubic polynomial.
In mathematical form Cubic Equation is given as:
A3x3 + A2x2 + A1x + A0 = 0,

Here, A3, A2, A1 and A0 are coefficients of x2, x2, x respectively. For solving cubic equation formula a closed-form formula is given which is known as cubic formula and it gives the solutions of a cubic equation. This formula is given as:
x = ∛[(-b3 / 27a3) + (bc / 6a2) – (d/ 2a) + {√(-b3/ 27a3) + (bc / 6a2) – (d/ 2a)}2 + {(c/3a – b2/9a2)}3 +∛ [(-b3 / 27a3) + (bc / 6a2) – (d/ 2a) + {√(-b3/ 27a3) + (bc / 6a2) – (d/ 2a)}2 + {(c/3a – b2/9a2)}3 - b/3a.
Using this formula one can easily solve different cubic equations by putting value of constants and variables in formula.