FACTORING POLYNOMIAL

Factoring polynomials is the opposite of multiplying polynomials by their factors. A polynomial of degree ‘n’ in variable x is an equation of the form axn + bxn-1 +cxn-2 +….+kx+ l, where each variable has a constant as its coefficient. A polynomial is an expression in which the addition or subtraction sign separates a combination of a constant and a variable.

MEANING:-

Factoring polynomials entails utilising prime factorization to decompose a given polynomial into a product of two or more polynomials. Polynomials can be factored to make them easier to simplify. To begin, write each term in the bigger statement as the product of its factors. To construct the required elements, the common factors across the phrases are removed in a second phase. We have the various ways for factoring polynomials, as well as some of the key concepts: remainder theorem, factor theorem, GCF, and long division.

PROCESS OF FACTORING POLYNOMIAL:-

A polynomial of the form axn + bxn – 1 + cxn – 2+………px + q can be factored using a variety of techniques, including grouping, employing identities, and substituting.

The exponent of x in this polynomial is n, and it has n components. The degree of the variable in the polynomial equation determines the number of components. To achieve the requisite factors, higher degree polynomials are reduced to simpler lower degree, linear or quadratic expressions.

The steps below will assist you in factoring polynomials. To factor a polynomial, follow the procedures outlined below.

Step1 :-Check to see if there is a factor that is common to all of the polynomial’s terms.

Step2 :-Decide on the best approach for factoring polynomials. To find the polynomial’s factors, you can utilise regrouping or algebraic identities.

Step3 :-Then Polynomial is written as the product of its factors

METHODS OF FACTORING POLYNOMIAL:-

Factoring polynomials can be done in a variety of ways depending on the expression. The degree of the polynomial, as well as the number of variables in the equation, define the factorization method. The four most common polynomial factoring methods are as follows:

1. Common factors methods

2. Grouping method

3. Factorising by splitting terms

4.  Factorising by algebraic expression

Method of common factor:-

The easiest method of factoring an algebraic equation is to take common factors of each of the terms of the presented expression. As a first step, the factors of each of the algebraic expression’s terms are written. The similar factors throughout the phrases are also taken into account when determining the possible elements. Applying the distributive property backwards is the same thing. Let us better understand this with the help of an example.

Suppose that we have a expression i.e. 3x²+9x

We can write the given expression as 3.x.x+3.3.x and the common factor is 3 and x and we can take it as common then the expression becomes 3x(x+3) i.e. the factorised expression.

Method of grouping:-

The approach of factoring polynomials by grouping is a step further than the method of identifying common factors. The goal here is to find groups from common factors in order to get the factors of a given polynomial equation. The number of terms in a polynomial equation is reduced to a lower number of groups. To determine the group of factors, we first separate each term of the supplied expression into its factors and then look for common words. With the help of the following example, let’s try to comprehend grouping for factoring.

Let us consider an expression : x² + 5x + 6

We can write the expression as x² + 3x + 2x + 6 = x.x + 3.x + 2.x + 6 = x(x+3) + 2(x+3) = (x+3)(x+2)

Hence, the factorisation is done by the grouping.

Method of splitting the terms:-

Factoring polynomials is a common method for solving quadratic equations. When factoring polynomials, the higher degree polynomial is frequently reduced to a quadratic equation. In order to obtain the components required for the higher degree polynomial, the quadratic equation must also be factorised.

A quadratic equation’s general form is x² + x(a + b) + ab = 0, which may be broken down into two parts (x + a)(x + b) = 0. Take the quadratic polynomial x² + x(a + b) + ab as an example.

=x.x + a.x + b.x + ab = x(x + a) + b(x + a) = (x + a)(x + b) = (x + a)(x + b) = (x + a)(x + b).

CONCLUSION:-

Factoring polynomials is the technique of expressing a polynomial as the product of its factors. Finding the values of the variables in a given expression or the zeros of a polynomial expression can be made easier by factoring polynomials. A polynomial of the form axn + bxn – 1 + cxn – 2+………px + q can be factored using a variety of techniques, including grouping, employing identities, and substituting.