How to Factor Polynomials?

Factoring Polynomials means the analysis of a given polynomial by the product of two or more polynomials using prime factoring. Factoring polynomials help in simplifying the polynomials easily. 

Related Topics

• How to Factor Trinomials

A step-by-step guide to factoring polynomials

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

Process of factoring polynomials

The following steps help with the polynomial factoring process. Follow the steps below to factorize a polynomial.

• If there is a common factor for all polynomial expressions, factor out.
• Determine the appropriate method for factoring polynomials. You can use regrouping or algebraic identities to find the factors of the polynomial.
• Write polynomials as the product of their factors.

Methods of factoring polynomials

The method of factorization depends on the degree of the polynomial and the number of variables in the expression. Here are four important ways to factorize polynomials:

grouping:

The grouping method for factoring polynomials is a further step in the method of finding common factors. Here our goal is to find groups of common factors to obtain given the factors of the given polynomial expression.

The number of terms of the polynomial expression is reduced to a lesser number of groups. First, we divide each given phrase into its factors and further aim at taking common terms to find the group of factors.

common factors:

This is the simplest way to factorize an algebraic expression by considering the common factors of the given expression. In the first step, the factors of each term are written in an algebraic expression.

In addition, the common factors across the terms are taken to obtain the possible factors. This is equivalent to using the distributive property in reverse.

algebraic identities:

The process of factoring polynomials can be easily done using algebraic identities. The given polynomial expressions show one of the algebraic identities. Also, sometimes the given phrase has to be modified so as to match the expression of the algebraic identities. Several algebraic identities are useful in factoring polynomials.

• \(\color{blue}{a^2\:-\:b^2\:=\:\left(a\:+\:b\right)\left(a\:-\:b\right)}\)
• \(\color{blue}{a^3\:-\:b^3\:=\:\left(a\:-\:b\right)\left(a^2\:+\:ab\:+\:b^2\right)}\)
• \(\color{blue}{a^3\:+\:b^3\:=\:\left(a\:+\:b\right)\left(a^2\:-\:ab\:+\:b^2\right)}\)
• \(\color{blue}{a^4\:-\:b^4\:=\:\left(a^2\:+\:b^2\right)\left(a\:+\:b\right)\left(a\:-\:b\right)}\)

splitting terms:

The process of factoring polynomials is often used for quadratic equations. While factorizing polynomials, we often reduce higher-order polynomials to a quadratic expression. In addition, the quadratic equation must be factorized to obtain the factors required for a higher degree polynomial.

The general form of a quadratic equation is \(x^2+\:x\left(a\:+\:b\right)\:+\:ab\:=\:0\) which can be split into two factors \((x + a)(x + b) = 0\). Consider the quadratic polynomial of the general form:

\(x^2+\:x\left(a\:+\:b\right)\:+\:ab\:=\:0\)

\(=x.x+ax+bx+ab\)

\(=x(x+a)+b(x+a)\)

\(=(x+a)(x+b)\)

Here in the above polynomial, the middle sentence is divided as the sum of two factors, and the constant term is expressed as the product of these two factors. Therefore, quadratic polynomials are expressed as the product of two expressions.

Factoring Polynomials – Example 1:

Factorize the polynomial \(6xy-4y+6-9x\). 

Solution:

To factorize polynomials, we see that we have no common factor among all terms. So regrouping them as \((6xy-4y)\) and \((6-9x)\).

\((6xy-4y) + (6-9x) = 2y(3x-2)- 3(3x-2)\)

\(= (3x – 2)(2y – 3)\)

Exercises for Factoring Polynomials

Factor each of the following polynomials.

1. \(\color{blue}{\:3x^2−2x+12x−8}\)
2. \(\color{blue}{3x^2-7x-6}\)
3. \(\color{blue}{4x^2+19x+12\:}\)
4. \(\color{blue}{6x^3-4x}\)
5. \(\color{blue}{x^3+64}\)

1. \(\color{blue}{\left(3x-2\right)\left(x+4\right)}\)
2. \(\color{blue}{\left(3x+2\right)\left(x-3\right)}\)
3. \(\color{blue}{(4x+3)(x+4)}\)
4. \(\color{blue}{2x\left(3x^2-2\right)}\)
5. \(\color{blue}{\left(x+4\right)\left(x^2-4x+16\right)}\)