Factorisation Of Polynomials

Factoring Polynomials means decomposing the given polynomial into a product of two or more polynomials using prime factorization. Factoring polynomials help in simplifying the polynomials easily.

What is Factoring of Polynomials?

The process of factoring polynomials involves expressing the polynomial as the product of its factors. Factoring polynomials help in finding the values of the variables of the given expression or to find the zeros of the polynomial expression.

A polynomial is of the form $ax^{n}+bx^{n-1}+cx^{n-2}+.....+px+q$ can be factorized using numerous methods: grouping, using identities and substituting.

In this polynomial, the exponent of $x$ is $n$ and it has $n$ factors. The number of factors is equal to the degree of the variable in the polynomial expression. Higher degree polynomials are reduced to a simpler lower degree, linear or quadratic expressions to obtain the required factors.

Process of Factoring Polynomials

The following are the steps for the process of factoring polynomials.

1. Factor out if there is a factor common to all the terms of the polynomial.
2. Identify the appropriate method for factoring polynomials.
3. Write polynomial as the product of its factors.

Methods of Factoring Polynomials

There are numerous methods of factoring polynomials, based on the expression. The method of factorization depends on the degree of the polynomial and the number of variables included in the expression. The four important methods of factoring polynomials are as follows.

• Method of Common Factors
• Grouping Method
• Factoring by splitting terms
• Factoring Using Algebraic Identities

Factoring Polynomials by Splitting Terms

The process of factoring polynomials is often used for quadratic equations. While factoring polynomials we often reduce the higher degree polynomial into a quadratic expression. Further, the quadratic equation has to be factorized to obtain the factors needed for the higher degree polynomial. The general form of a quadratic equation is $x^{2}+x(a+b)+ab=0$ which can be split into two factors $(x+a)(x+b)=0$

$x^{2}+x(a+b)+ab$

$=x.x+ax+bx+ab$

$=x(x+a)+b(x+a)$

$=(x+a)(x+b)$

In the above polynomial, the middle term is split as the sum of two factors, and the constant term is expressed as the product of these two factors. Thus the given quadratic polynomial is expressed as the product of two expressions.

Example: Let us understand this better, by factoring a quadratic polynomial $x^{2}+7x+12$

$x^{2}+7x+12$

here the middle term is 7 and last term is 12 . the possible combinations of splitting the middle term such that the product of factors of middle term and last term matches is shown in the below table.

Middle term	Factor 1	Factor 2	product of factor 1 and factor 2	Last term	Is Product of factor 1& 2 = Last term
7	1	6	1 x 6 =6	12	No
7	2	5	2 x 5 =10	12	No
7	3	4	3 x 4 =12	12	Yes

$x.x+3x+4x+3.4$

$x(x+3)+4(x+3)$

$(x+3)(x+4)$