# 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,

- Factorization Method
- Method of Quadratic formula
- 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 :

## 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)±(b
^{2}– 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 :

## 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 :

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