Factorisation Using Division

What is Factorization?

Let us first learn what is factorization? Factorization is nothing but the process of minimizing the bracket of a quadratic equation, rather than expanding the bracket and transforming the equation to a product of factors which cannot be reduced further. For example, factorising (x²+ 5x+ 6) brings forth the result as (x+2) (x+3). Here, (x+2) (x+3) denotes a factorization of a polynomial (x²+5x+6). These factors can either be algebraic expressions, variables, or integers. Basically, factorisation is the reverse function of multiplication. A form of decomposition,factorization brings forth the gradual splitting of a polynomial into its factors. In this article, we will discuss what is factorization, factorization using division, factorization by division, finding factors by long division,etc.


Factorization Definition

Factorisation or factoring is defined as the splitting or disintegration of a unit (for instance a number, a matrix, or a polynomial) into a product of another unit or factors, which when multiplied together obtains the original number or a matrix, etc. 

It is simply the transformation of an integer or polynomial into factors such that when two integers multiplied together obtains the result as the original integer or polynomial. In the factorization method, we minimise any algebraic or quadratic equation into its simpler form, where the equations are expressed as the product of factors rather than expanding the brackets. The factors of any equation can either be an integer, a variable, or an algebraic expression itself


Factorization Using Division Important Points

Following points should be noted while factoring the polynomial using the division method.

  • Factorization by division method is a simpler and traditional approach of finding the factors of a polynomial expression.

  • Determining factors of a polynomial is simply like performing any simple division, the only thing should be considered is the accuracy of variables and coefficients.

  • There are two ways through which factorization of polynomials can be done. The first method is the simple division method and the second method is the long division method.


Finding Factors: The long Division Way Introduction

The following are the steps to be followed for finding factors by the long division way.

  1. We will first arrange the given polynomials in descending order. We will replace every missing term with 0.

  2. In the second step, we will divide the first term of the dividend by the first term of the divisor. With this, we will get the first term of the quotient.

  3. Next, we have to multiply the divisor by the first term of the quotient.

  4. Further, the next term will be brought down by subtracting the product from the dividend. The next term which is brought down and the difference of the product and dividend will be the new dividend.

  5. Repeat step 2 and 4 to determine the second term of the quotient.

  6. Continue the process till we get a reminder. The remainder can be either zero or lower than the divisor.

We will get the value of the remainder equals to zero if the divisor is a factor of dividend. We will get the remainder lower than the divisor if the value of the remainder is not equal to zero.

Factorization By Division 

The first step is to split the polynomials into its direct factors in the factorization by simple division method. For example if we divide, 8z³ + 7z² +6z by 2, we will split the given equation in its basic factors i.e.2x(4z)² + 2x(7/2 × z) + 2z(3).

In the next step, we will write the common factors separately i.e. 2z{(4z)² +(7/2 × z) + (3)}/2z.

In the last step, we will divide the given expression as mentioned in the question i.e  2z{(4z)² +(7/2 × z) + (3)}.

Hence, the answer will be: 4z² +(7/2z) + 3.

Factorization By Division Example

Factorise the below by simple division method:

Divide:  (p2qr+ pq2r+ pqr2) by 4pqr

Solution: 2 ×2× 2× 2 [( p × p × r × r) + ( p × q× q ÷ r) +( p ×q × r × r)]

16pqr (p + q+ r)

Now, we will divide the polynomial as asked in the question

= 4×4 pqr (p + q+ r)/ 4pqr

= 4(p+q+r)


Solved Examples

1. Factorise the Following By Division Method

Divide: 3x3 + 4x + 11 x2  - 3x + 2

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2. Divide: 3x3 + 4x + 11 x3 - 3x + 2

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Step 11 : 3x² - 5x + 6


1. Factorization of 4x-20 Gives

  1. 2x-4

  2. 4(x-5)

  3. 5(x-4)

  4. 20x

2.  Factorization of p+ pq + 2q+ 2q²

  1. p² + 2pq

  2. p+ q

  3. (1+q)(p+2q)

  4. None of the above

FAQ (Frequently Asked Questions)

1. Define Polynomial and Mention its Types

A polynomial is defined as an algebraic expression that includes two or more than two terms. The word polynomial is derived from two words poly meaning "many" and nominal means "number".The polynomial includes the following.

Constant such as 4,5,6,7, etc

Exponents such as 3 in x

Variable such as x,y,z etc

There are three different types of polynomials. These are

Monomial -It is an algebraic expression that includes only one term. For example, 4x, 3x etc

Binomial- It is an algebraic expression that includes two terms. For example, 5x+3, 2y +6 etc.

Trinomial- It is an algebraic expression that includes more than two terms. For example, 2x+3+5, -6a +2x +7.

2. Explain Factorization of Polynomials.

Factorization is basically a decomposition of an expression into a product of its factors.The process involved in splitting a polynomial into the product of its factors is considered as the factorization of polynomials.For example, let us factorize the below polynomial

x³ + 3x² - 6x - 18 = 0

(x³ + 3x² ) can be splitted as x² (x+3) whereas 6x-18 can be splitted as -6(x + 3)

Factorization often converts an expression into a form that can be algebraically manipulated easily and also has solutions that can be identified easily.Factorization also brings forth the precise defined relationships.