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Introduction to Factoring

The process of finding factors is known as factoring. This process can also be called factorization. It can also be defined as, factoring consists of a number or any other mathematical object as the product of two or more factors. For example, 3 and 5  are the factors of integer 15.


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Factoring Algebra

Factoring algebra is the process of factoring algebraic terms. To understand it in a simple way, it is like splitting an expression into a multiplication of simpler expressions known as factoring expression example: 2y + 6 = 2(y + 3). Factoring can be understood as the opposite to the expanding. Different types of factoring algebra are given below so that you can learn about factoring in brief.


Types of Factoring Algebra

Different types of factoring algebra are discussed below:

  • Factoring out the Greatest Common factor.

  • The sum-product pattern.

  • The grouping pattern.

  • Perfect square trinomials.

  • Difference of Squares.

Let us discuss the basic two methods of factoring which is used frequently to factorise the polynomial. The most popular formula used to find the factors of a polynomial in the Quadratic equation is Shridhar’s formula.


\[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\]


Greatest Common Factor

In this, we have to find the greatest common factor of the given polynomial to factorise it. This process is a type of reverse procedure of distributive law.

Distributive law p(q + r) = pq + pr

In the case of factorisation, it is opposite of distributive law

pq + pr = p(q + r)

Where ‘p’ is the greatest common factor of the given polynomial.


Factorisation Problems:

1) Factorise 6x2 + 3x

Now, take the common multiple out from the above polynomial

3 is the common multiple for the given problem.

= 3x (2x + 1)

Therefore, 6x2 + 3x = 3x(2x + 1)


2) Factorise 2x2 + 8x

Here, 2 is the common multiple for the given problem 

= 2x(x + 4)

Therefor, 2x2 + 8x = 2x(x + 4)

Note: This method is applicable if each term in the polynomial shares a common factor.


Factoring Polynomial by Grouping

In this method, the given polynomial is grouped in the pairs to find the zeros. This method is also called factoring by pairs.

Factorisation examples: Factorise x2 - 15x + 50


To solve the problem by grouping, find the two numbers which when added gives -15 and multiplication gives 50.

So, -5 and -10 are two numbers

Like, (-5) × (-10) = 50

(-5) + (-10) = -15

Therefore, The given polynomial can be written as:

X2 - 5x - 10x + 50 = 0

x(x - 5) - 10(x - 5) = 0

In this taking (x - 5) as common factor;

We get, (x - 5)(x - 10)

Hence, The factors are (x - 5) and (x - 10).

Note: This method is applicable if the polynomial of the form x2 + bx + c and there are factors of ac that add up to b.


Factoring Rules:

Some of the basic rules of the factoring are:

  • If the third co-efficient(c) is "plus", then the factors will be either both "plus" or else both "minus".

  • If the second coefficient(b) is "plus", then the factors are both "plus".

  • If the second coefficient(b) is "minus", then the factors are both "minus".

  • In either case, look for factors that add to b.

  • If the third coefficient(c) is "minus", then the factors will be of alternating signs; that is, one will be "plus" and one will be "minus".

  • If the second coefficient(b) is "plus", then the larger of the two factors is "plus".

  • If the second coefficient(b) is "minus", then the larger of the two factors is "minus".

FAQs (Frequently Asked Questions)

1. How to Factor Algebraic Equations?

Answer. In maths, Factoring means finding numbers or expressions that multiply to form the given number or expressions. Factoring is very essential when dealing with Quadratic equations and polynomials. To factor numbers and basic algebraic expressions. Steps to be followed are: understand the definition of factoring, understand that expressions can also be factors, and finally apply the distributive property of multiplication to factor algebraic equations.

2. Where to Use Factoring in Real Life?

Answer. Factoring is used in the real-life but we do not realize. Some of the uses are: Dividing something equally, Factoring with money, Comparing prices, Understanding time, and travelling with factors are the real-life uses.

3. What is the Smallest Factor of any Number?

Answer. 1, is the smallest factor of any number. All the numbers will have two factors One is 1 and the other is number itself. The number having only two factors 1 and the number itself is also called prime numbers. Some of the examples are 7 × 1 = 7, 3 × 1 = 3.

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