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Finding Roots of a Polynomial Equation of Any Arbitrary Degree

1. A Polynomial Equation of Degree \(N\) in 1 Variable is given as follows
   
   \(A_nx^{n} + A_{n-1}x^{n-1} + A_{n-2}x^{n-2} + A_{n-3}x^{n-3} + A_{n-4}x^{n-4} + \cdots + A_{2}x^2 + A_{1}x + A_0=0\)   ...(1)
   
   where the Co-efficient of \(x^{n}\) Cannot be Zero i.e. \(A_n\neq0\)
2. A Single Root for any Polynomial Equation of Degree \(N\) (where \(N \geq 2\)) of 1 Variable can be found out using Newton-Raphson Method as follows
   1. Calculate the Derivative Polynomial of the given Polynomial. For example if \(P(x)\) is the Polynomial Equation of Degree \(N\) of Variable \(x\), find it's Derivative Polynomial Equation \(P'(x)\) of Degree \(N-1\) .
   2. Select any Arbitrary Value \(V\) as Initial Value for Variable \(x\) and repeatedly evaluate the following expression
      
      \(V = V - {\Large \frac{P(V)}{P'(V)}}\)
      
      where \(P(V)\) and \(P'(V)\) are the Values of the Polynomials \(P(x)\) and \(P'(x)\) respectively for \(x=V\), till the Value of \(V\) Converges to a Root of the Polynomial \(P(x)\) upto the required precision.
3. Any Polynomial Equation of Degree \(N\) (where \(N \geq 2\)) in 1 Variable always has N Roots/Solutions that can be calculated recursively using the following steps
   1. Find a Single Root \(R\) of the Polynomial Equation of Degree \(N\) using Newton-Raphson Method given above and Add it to the List of Roots.
   2. If the Degree of Polynomial Equation is 2 (i.e. \(N = 2\)), then equation (1) can be written as
      
      \(A_2x^2 + A_1x + A_0=0\)   ...(2)
      
      If \(R\) is One of the Calculated Roots of the Polynomial Equation (2) above, then it can be can be written as a Product of 2 Polynomials as follows
      
      \((x-R)(B_1x + B_0) = 0\)   ...(3)
      
      \(B_1x^2 + (B_0 -B_1R)x + B_0 = 0\)   ...(4)
      
      Using equation (3) above, the Other Root can be calculated as
      
      \(x= -{\Large \frac{B_0}{B_1}}\)   ...(5)
      
      Now, by comparing equations (2) and (4) above we get that
      
      \(B_1=A_2\) and \(B_0-B_{1}R=A_{1}\hspace{3mm}\Rightarrow B_0=A_{1} + B_{1}R \hspace{3mm}\Rightarrow B_{0} = A_{1} + A_{2}R\)   ...(6)
      
      Now, by substituting the value of \(B_0\) and \(B_1\) from equation (6) in equation (5), the Other Root can be calculated as
      
      \(x= -{\Large \frac{A_{1} + A_{2}R}{A_2}}\)   ...(7)
   3. If the Degree of Polynomial Equation is Greater than 2 (i.e. \(N > 2\)), and if \(R\) is One of it's Calculated Roots, the the Polynomial Equation of Degree \(N\) can be written as a Product of 2 Polynomials, a Polynomial of Degree 1 and a Polynomial of Degree \(N-1\) as follows
      
      \((x-R)(B_{n-1}x^{n-1} + B_{n-2}x^{n-2} + B_{n-3}x^{n-3} + B_{n-4}x^{n-4} + \cdots + B_{2}x^2 + B_{1}x + B_0)=0\)   ...(8)
      
      \(\Rightarrow B_{n-1}x^{n} + (B_{n-2} - B_{n-1}R) x^{n-1} + (B_{n-3} - B_{n-2}R) x^{n-2} + (B_{n-4} - B_{n-3}R) x^{n-3} + \cdots + (B_1 - B_{2}R )x^2 + (B_0 - B_{1}R )x - B_0R=0\)   ...(9)
      
      Since the Polynomial Equation with Degree \(N\) (as signfied by equation (1) above) and Polynomial Equation with Degree \(N-1\) (as signfied by equation (9) above) are same, therefore the values of Coefficients of Polynomial Equation with Degree \(N-1\) (i.e \(B_{n-1}, B_{n-1}, B_{n-3}, \cdots, B_{2}, B_{1}, B_{0}\)) can be found out in terms of Coefficients of Polynomial Equation with Degree \(N\) (i.e \(A_{n}, A_{n-1}, A_{n-2}, A_{n-3}, \cdots, A_{2}, A_{1}, A_{0}\)) as follows
      
      \(B_{n-1} = A_n\)
      
      \(B_{n-2} - B_{n-1}R = A_{n-1}\hspace{3mm}\Rightarrow B_{n-2} = A_{n-1} + B_{n-1}R\)
      
      \(B_{n-3} - B_{n-2}R = A_{n-2}\hspace{3mm}\Rightarrow B_{n-3} = A_{n-2} + B_{n-2}R\)
      
      \(B_{n-4} - B_{n-3}R = A_{n-3}\hspace{3mm}\Rightarrow B_{n-4} = A_{n-3} + B_{n-3}R\)
      
      \(\vdots\)
      
      \(B_{1} - B_{2}R = A_{2}\hspace{3mm}\Rightarrow B_{1} = A_{2} + B_{2}R\)
      
      \(B_{0} - B_{1}R = A_{1}\hspace{3mm}\Rightarrow B_{0} = A_{1} + B_{1}R\)   ...(10)
      
      Once the Coefficients of Polynomial Equation with Degree \(N-1\) (i.e \(B_{n-1}, B_{n-1}, B_{n-3}, \cdots, B_{2}, B_{1}, B_{0}\)) have been found, set the Polynomial Equation of Degree \(N-1\) as given in equation (8) above as the New Polynomial Equation of Degree \(N\) and continue to Find Other Roots by going back to Step a.
4. You can use the Polynomial Roots/Factors Calculator to find the Roots/Factors of Polynomial Equations/Expression.