Factorisation

When we break a number or a polynomial into a product of many factors of other polynomials, which, when multiplied, gives the original number, it is called factorisation. 

To factorise a number, use the factorisation formula. The factorisation is the process of converting one entity (for example, a number, a matrix, or a polynomial) into a product of another entity, or factors, which, when multiplied together, yield the original number.

The factorisation formula divides a large number into smaller numbers, known as factors. A factor is a number that divides a given integer completely without leaving any remainder.

For example -  Prime Factorisation of 28 = 2 X 2 X 7 and 

Before starting factorisation, let us first discuss an important mathematical term, ‘Factor’.

What is a Factor? 

Factors are the numbers, algebraic variables or an algebraic expression which divides the number or an algebraic expression without leaving any remainder.

For example factor of 9 is 1,3,9

Algebraic Factorization

Because they divide 12 without leaving a remainder, the numbers 1, 2, 6, and 12 are all factors of 12. It is a fundamental algebraic procedure for simplifying expressions, fractions, and solving equations. Algebra factorization is another name for it.

The factors of an algebraic expression 10xyz = 2 × 5 × x × y × z.

Similarly,

The algebraic expression 5(x + 1) (z + 2) can be written in irreducible factor form as:

5(x + 1) (z + 2) = 5 × (x + 1) × (z + 2)

What is the Definition of a Term?

In an expression, it's anything that has to be added or removed (subtracting means adding a negative quantity).

For example- 2a+9, here 2a and 5 are terms for this equation. 

Methods of Factorization

The algebraic expressions can be factored in using one of four approaches are as follows. 

1. Method based on common factors- List the prime factors of every equation. Prime factors are those numbers that are prime numbers. Circle each common factor — that's, each factor that is an element of each variety within the set. Multiply all the circled numbers. The result, in the end, is the common factor.

Step 1: Each term of a given algebraic expression is written as a product of irreducible factors.

Step 2: The common factors are taken out and the rest of the expression is combined in the brackets.

For example 1: Factorize 6xy + 15yz

Step 1:  we have, 6xy = 2 × 3 × x × y

                            15yz = 3 × 5 × y × z

Step 2: Common factors of these terms are 3 and y.

Therefore, 6xy + 15yz = (2 × 3 × x × y) + (3 × 5 × y × z)

                  6xy + 15yz = 3y (2x + 5z)

2. Method of reorganizing terms- Regrouping allows us to rearrange the expression's terms, which leads to factorization.

When we do this, a common factor comes out from all the groups and leads to the required factorisation of the expression. 

For example 2: Factorize 3x²y - 9x + 4xy – 12

Since, there is no common factor among the terms of given expression. So, we will group them separately to get a common factor from these groups.

We observe that the terms 3x²y - 9x have a common factor of 3x and the terms 4xy – 12 have a common factor of 4.

i.e.   3x²y - 9x = 3x (xy - 3) and 4xy – 12 = 4 (xy – 3)

Putting these grouped terms together, we get:

3x²y - 9x + 4xy – 12 = 3x (xy – 3) + 4 (xy – 3)

                            = (3x + 4) (xy – 3)

Therefore, the factors of 3x²y - 9x + 4xy – 12 are (3x + 4) and (xy – 3).

3. Identity-based factorization- There are some identities that can be used to make factorization much easier. A number of expressions to factorize are or can be written as: a² + 2ab + b²,a² - 2ab + b²,a² - b² and x² + x(a+b )+ ab

For example3: Factorize x² + 10x + 25

Above algebraic expression x² + 10x + 25 matches with the form of identity a² + 2ab + b²

Where, a = x and b = y

Such that, a² + 2ab + b² = x²+ 2(x)(y) + 5²

               x²+ 2(x)(y) + 5² = (x + 5)²  

Therefore, the factors of x² + 10x + 25 are (x + 5) and (x + 5).

For example 4: Factorize 9x²– 25y²

Above algebraic expression 9x²– 25y² matches with the form of identity a² - b² = (a + b) (a - b)

Where, a = 3x and b = 5y

Such that, a² - b²  = (3x)² – (5y)²     

                  9x²– 25y² = (3x)² – (5y)²

                                   = (3x + 5y) (3x – 5y)

Therefore, the factors of 9x²– 25y² are (3x + 5y) and (3x – 5y).

4. Method of differences in squares of 2 numbers. -  Polynomials reflect a difference of squares because the two squares are subtracted. The difference of squares pattern can be used.

For example,x²-25  can be factored as (x+5) (x-5). The pattern (a+b)(a-b)=a²-b² is used in this procedure, which may be checked by expanding the parenthesis in (a+b) (a-b).

5. Factoring algebraic expression of form x² + px + q

For factoring an algebraic expression of the form x² + px + q, we find two factors a and b of the constant term i.e. q such that:

Product of a and b i.e. ab = q 

And sum of a and b i.e.  a + b = p

Then, the given algebraic expression becomes:

  x² + (a + b) x + ab 

⇒ x² + ax + bx + ab 

⇒ x (x + a) + b (x + a) 

⇒ (x + a) (x + b)  which are the required factors of given algebraic expression.

For example 5: Factorize x² + 8x + 15

 First of all, we will factorize the constant term 15 into two factors such that 

 their Sum = 8 and 

 Product = 15

Clearly, the numbers are 5 and 3

So we can write, x² + 8x + 15 = x² + 5x + 3x + 15

                                                   = x (x + 5) + 3(x + 5)

                                                   = (x + 5) (x + 3)

Therefore, the factors of x² + 8x + 15 are (x + 5) and (x + 3).

What is the Purpose of Factoring?

Factoring is a typical mathematics procedure for separating the components (or numbers) that multiply to generate a new number. There are several components in certain integers. When you multiply the factors of 6 and 4, 8 and 3, 12 and 2, and 24 and 1, you get the number 24. Factoring can be used to solve a variety of math problems.

When solving quadratic polynomials, factoring formulas algebra is especially important. When reducing formulas, we usually have to remove all of the brackets, but in some cases, such as with fractional formulas, we can use factorisation to shorten the formula.

You already possess all of the skills required to factor if you understand the fundamentals of multiplication and division. The factors of a number are any numbers that can be multiplied to produce that number. A number can also be factored in by dividing it repeatedly. While factoring in huge numbers can be challenging at first, there are a few easy strategies you can learn to discover the factors quickly.

Finding the factors of a number includes finding all of the terms that multiply together to produce that number. Reduce a number to its factors, which are prime numbers, to factor it entirely.

What is a Factor Tree?

A factoring tree can be used to represent the division of a big integer into its prime components. Place the number to be factored at the top of the statement and divide it by its factors into steps. Place the two factors of a number below each time you divide it. Divide until all of the numbers are reduced to their prime factors.

After you've mastered the fundamentals of factoring, you may be asked to find the greatest common factor of two integers or expressions. Create a list of both numbers' factors to identify the greatest common factor. The highest number that appears on both lists is the greatest common factor.

Purpose of a Factor in Real Life

1. Another typical operation that relies on factoring is money exchange. In principles of factoring, the two factors of 100 are 4 and 25.

2. Factoring is your gateway to doing great things in life. If you want to be a pharmacist or astronomer or biologist or physicist, or a programmer or be your manager and be competitive, or do anything more than 9-to-5 work; if you want to be a leader in your field and do great things, you will need mathematical skills built on college algebra, and therefore you need to learn about input.

The Importance of Mathematics

Mathematics is the study of measurements, numbers, and space, and it was one of the first sciences to be developed due to its importance and value. The term "mathematics" is derived from Greek and means "learning proclivities," and there are many branches of mathematics in science that deal with numbers, such as geometric shapes, algebra, and others.

Mathematics is used in almost every aspect of life, from daily tasks like timekeeping, driving, and cooking to professions like accounting, finance, banking, engineering, and software development. These responsibilities necessitate a solid mathematical foundation, and scientific experiments necessitate mathematical techniques. They are a language used to describe scientists' work and accomplishments.

Mathematical reasoning is aided by analytic thinking also.

1. Mathematics aids in the development of critical thinking skills.

2. It aids in the explanation of how things work.

3. It aids in the development of wisdom.

4. It boosts the speed with which one's intuition works.

5. It aids in the development of the child's intellect.

6. When mathematics is applied as a career, money can be made.

7. It is critical in a world that is continuously changing.

8. It allows the child to interact with the rest of the world.