Algebraic Methods for Polynomial roots

This post talks about the algebraic methods for finding roots of polynomials. (or Solutions of a polynomial) . Here, we’ll discuss- Factorization Method , Method of Quadratic formula & the Synthetic Division Method.

As we know, the algebraic polynomials are the functions of some variable (say , ‘x’) with a series of ascending powers of that variable.  and as a matter of fact, the highest power of the polynomial, denotes the number of roots of a polynomial.

There are various methods of finding roots of the polynomials , such as, Algebraic methods and the Numerical methods. Here, we are discussing the Algebraic methods only.

Algebraic Methods for Polynomial Roots  :

there are three algebraic methods to find the root. they are,

  1. Factorization Method
  2. Method of Quadratic formula
  3. Synthetic division method

These are the algebraic methods, Firstly, we discuss Factorization method, further we’ll discuss accordingly.

 Factorization Method :

This algebraic method for finding root of polynomial is suitable only for the Quadratic equations , In other words, the equations with highest power 2 .

  • The coefficient of the square term should be unity, not compulsorily.
  • Factorize the middle term so that the addition of factors gives the middle term & their product gives the product of coefficients of the square term & the constant.
  • The factors, thus obtained gives the root of the equation.

Let us consider an example below :

Factoor Method example


Method of Quadratic formula :

This algebraic method for finding root of polynomial is  also suitable only for the Quadratic equations , In other words, the equations with highest power 2 .

  • Denote the coefficients of the square term by- ‘a’ , the coefficient of variable term by – ‘b’ & the constant term by – ‘c’.
  • Now, use the formula :- ((-b)±(b2 – 4ac)) /(2a) , this formula gives the pair of solution for the given equation.
  • This is the solution of given quadratic equation.

Consider the example below :

Quadratic formula method


Synthetic Division method :

This is the algebraic method for finding root of polynomial is suitable for any polynomial. This method also works with the coefficients of the terms of the polynomial.

  • Firstly, we have to guess one of the roots of the equation, then using this root , we find remaining roots.
  • Write the coefficients of the given equation at right side of the line in division table, & write the root in step 1 to left side of the line.
  • Multiply the initial coefficient by the root &  subtract it from the second coefficient, repeat this until the last answer comes out to be zero.
  • The numbers at the bottom of the division table is the quadratic equation, apply one of the two methods mentioned above, and get the roots.
  • This gives the complete set of roots of the equations.

Consider the example below :

Synthetic Method


Above are the algebraic for finding root of polynomials. Further Numerical Methods of finding roots we’ll discuss in later posts…