Factorisation of Polynomials

When a polynomial is written as a product of polynomials, each of the polynomials in the product is a factor of the original polynomial.

Representation of the polynomials as the product of their factors is called factorisation.

If p(x) and g(x) are two polynomials, such that the degree of p(x) is greater than or equal to degree of g(x) and g(x) is not equal to 0,then there exists polynomials q(x) and r(x) such that

p(x) = g(x).q(x) + r(x), where r(x) = 0 or the degree of r(x) < the degree of g(x).

Remainder theorem states that if p(x) be any polynomial of degree greater than or equal to one and a be any real number, then if p(x) is divided by a linear polynomial x – a, the remainder obtained is p(a).

Factor theorem states that if p(x) be any polynomial of degree greater than or equal to one and a be any real number, then

    x – a is a factor of p(x), if p(a) = 0, and

    p(a) = 0, if x – a is a factor of p(x).

Following are the methods of factorisation:

    By common factors

    By grouping the terms

    By splitting middle term

    By difference of two squares

    By using identities

    By sum or difference of two cubes

    By using long division method on cubic polynomials

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