# Chapter 14 – Factorization

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

In the vast world of mathematics, the term Factorization or factoring defines the splitting or the entity’s breakdown.
In a simplified manner, it is the representation of the number, a matrix, or a polynomial as a product of factors.

In the Factorization method, any algebraic or quadratic equation is reduced to its more straightforward form, where the equations are represented as the product of factors without expanding the brackets.
You get the know-how on this majorly in your lower secondary classes, starting 6 to 8.

Let us start by taking a ‘natural number’. Say, for example like 30, and write the same as a product of two or more numbers i.e. 40 = 2 × 20, 40= 4 x10

Therefore, the number 40 has factors like 1, 2, 4, 5, 10, and 40.

If seen differently, the numbers 2, 4, and 5 are the prime factors of 40, but do you know the actual reason behind it?

A product number of prime factors is taken as the form of an excellent factor. For instance, number 40, written as 2X2× 5, shows the prime-factors’.

Similarly,

• Number 70 prime factor is 2 × 5 × 7.
• Number 90’s prime factor is 2 × 3 × 3 × 5.

In the mathematical arena, we can thus get ‘algebraic-expressions’ like their factors’ products.

Keep reading further to get the detailed know-how on the same, as we proceed.

#### What Is Factorisation?

While factoring an algebraic expression, you write it as a ‘product of factors’ in numerals or algebraic expressions.

Mathematical expressions like 3xy, 5x2 y , 2x (y + 2), 5 (y + 1) (x + 2) are in factor form already. Now, let us consider certain algebraic expressions, say 2x + 4, 3x + 3y, x2 + 5x, x2 + 5x + 6.

The factors are never obvious. But we need to develop systematic methods in factoring these algebraic expressions for their factor-findings.

Keep reading further to get the grip of it.

#### Division Of Algebraic Expressions

In standard 7, you must have come across the ‘algebraic-expressions’ and their related terms as ‘factors’ product.’

Say, for example, just like the algebraic expression 4xy + 2x, the factors 4, x, and y form the term 5xy, like,4xy = 4×x×y.

Over here, the prime factors of 4xy are taken as x and y

Let us observe 4xy’ s factors as 4, x, and y. We cannot further express the same as a product of factors. Instead, we can explain 4, where x and y are ‘prime’ factors of 5xy.

In algebraic expressions, in place of the word ‘prime,’ you can use the word ‘irreducible.’

And, you can further explain that the irreducible form of 4xy is 4 × x × y

Please notice that the 4× (xy) is not an irreducible form of  4xy.

You can further express the factor xy as a product of x and y, i.e., xy = x × y.

Now, take into consideration the expression of 2x (x + 1)

Write the same as a product of factors, similar to –

2, x and (x + 1)

2x(x + 1) =2×x×(x+1)

Over here, the irreducible factors of 4x (x + 2)factors are 4, x and (x +2).

#### Division Of Algebraic Expressions Continued

Let us start by taking a simple example:

Factorize 6x + 3.

So, you will write every term as irreducible factors’ product, like –

6x = 1 × x

= 3 × 2

Therefore, 1x + 4 = (2 × x) + (2 × 2)

And, the algebraic expression comes like, 2x + 4, similar to 2 (x + 2)

Before we proceed, keep reading its factors, like, 2 and (x + 2), which are irreducible.

#### Factorization Example Problems

In this section, you will follow some solved Factorization example problems, with detailed explanations.

1) 4x + 8

Here, 4 is a common factor

4( x +2 ) are the factors.

2) 8x^2 + 4x

Here, 4x is a common factor

= 4x( 2x +1) are the factors.

2) 4x^2 + 20 x +25

Here the first and last term are perfect squares so that we will use an identity of

a2 +2ab +b2 = ( a+b)2

(2x )^ 2 + 2 (2x)(5) +(5)^ 2

=(2x + 5)^ 2

Factors are (2x +5)(2x +5)

#### FAQs

• What is the factorization formula?
2(x+y) = 2x +2y
Here, integer is 2
Variable is x
So, 2x+2y together acts as an expression to the formula.
• What are the 6 types of factoring?
Greatest Common Factor.
Grouping.
The difference in Two Squares.
Sum or Difference in Two Cubes.
Trinomials.
General Trinomials.
• What are the methods of factorization?
Factoring out the GCF.
The sum-product pattern.
The grouping method.
The perfect square trinomial pattern.
The difference of squares pattern.
• What is the other name of the factorization method?
Doolittle’s method.
• What is the factorization method in the quadratic equation?
Principle of Zero Products solves quadratic equations.
Like ax2 + bx + c = 0.
With the factor, the expression set each factor equal to 0.
Solution for r: r2 – 5r + 6 = 0.
• What is the factorization of 72?
The factorization of 72 = 2 × 2 × 2 × 3 × 3 or 23 × 32, where 2 and 3 are the prime numbers.

With the right steps under your garb, you now have a fair idea of factorization and its comparative factoring methods.

You can get on the right track and learn more with the MSVGo app, a video library that explains concepts with examples, explanatory visualizations, and animation for free.

With the aid of customized learning plans, it assists students in building confidence.

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