Factorisation

In Mathematics, factorisation or factoring is defined as the breaking or decomposition of an entity (for example a number, a matrix, or a polynomial) into a product of another entity, or factors, which when multiplied together give the original number or a matrix, etc. This concept you will learn majorly in your lower secondary classes from 6 to 8.

Example: Factorisation of x2 – 4 is (x – 2) (x + 2). It means both (x- 2) and (x + 2) are the factors of x2 – 4.

It is simply the resolution of an integer or polynomial into factors such that when multiplied together they will result in original or initial the integer or polynomial. In the factorisation method, we reduce any algebraic or quadratic equation into its simpler form, where the equations are represented as the product of factors instead of expanding the brackets. The factors of any equation can be an integer, a variable or an algebraic expression itself.

What is Factorisation in Mathematics?

Factorisation of an algebraic expression means writing the given expression as a product of its factors. These factors can be numbers, variables, or an algebraic expression.

To the factor, a number means to break it up into numbers that can be multiplied to get the original number. For example,

 24 = 4 × 6 4 and 6 are the factors of 24 9 = 3 × 3 3 is the factor of 9

Also, numbers can be factorized into different combinations. There are different ways to find the Factors of a Number.  To find the factors of an integer is an easy method but to find the factors of algebraic equations is not that easy. So let us learn to find the factors of quadratic polynomial.

Factorisation in Algebra

The numbers 1, 2, 6, and 12 are all factors of 12 because they divide 12 without a remainder. It is an important process in algebra which is used to simplify expressions, simplify fractions, and solve equations. It is also called as Algebra factorization.

Terms and Factors

What is a Term?

It is something which is to be added or subtracted (subtracting is adding a negative number) in an expression.

If 2x + 7 is an expression, then 2x and 7 are the terms.

Sum = term + term

What is a Factor?

It is something that is to be multiplied in an expression.

Product = factor × factor

For example :

p = 4(2q – 6)

4 and 2q – 6 are the factors whereas 2q and 6 are the terms.

Basic Factorisation Formula

• a2 – b2 = (a – b)(a + b)
• (a + b)2 = a2 + 2ab + b2
• (a – b)2 = a2 – 2ab + b2
• a3 – b3 = (a – b)(a2 + ab + b2)
• a3 + b3 = (a + b)(a2 – ab + b2)
• (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
• (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca

Factorisation Methods

There are four methods to factorise the algebraic expressions.

• Common factors method
• Regrouping terms method
• Factorisation using identities
• Factors of the form (x+a) (x+b)

Method of Common Factors

In this method, we simply take out the common factors among each term of the given expression.

Example: Factorise 3x + 9.

Since, 3 is the common factor for both the terms 3x and 9, thus taking 3 as a common factor we get;

3x + 9 = 3(x+3).

Regrouping of Terms Method

Regrouping means rearranging the given expression based on the like terms or similar terms.

For example, 2xy + 3x + 2y + 3 can be rearranged as:

2xy + 3x + 2y + 3

Expanding the terms into factor form.

= 2 × x × y + 3 × x + 2 × y + 3

Rearrange to get the common factor

= x × (2y + 3) + 1 × (2y + 3)

Now (2y + 3) is the common factor we can take out.

= (2y + 3) (x + 1)

Thus, these are the required factors.

Factorisation using identities

By using the common identities, we can factorise the given expression.

Example: Factorise 4x2 – 9.

Solution: By using the algebraic identities, we know;

a2 – b2 = (a – b) (a+b)

Hence, we can write,

4x2 – 9

= (2x)2 – 32

= (2x + 3) (2x – 3)

Note: For the rest identities check with the formulas for factorisation mentioned in the above section.

Factors of the Form (x+a) (x+b)

If a given expression is in the form of x2 + (a + b) x + ab, then the factors will be (x+a) and (x + b).

Example: Factorise x2 + 5x + 6.

Solution: If we compare the given expression with x2 + (a + b) x + ab, then;

a+b = 5   ….(i)

ab = 6  ….(ii)

Now we will try putting the values for a and b such that they satisfy the above two equations.

If we put a = 4 and b = 1, then a + b = 5 satisfies the eq.(i) but a.b = 4, does not satisfy eq.(ii).

Again, if a = 2 and b = 3, then a + b = 5 satisfies eq.(i) but a.b = 6, does not satisfy eq.(ii).

x2 + 5x + 6

= x2 + (2 + 3)x + 2.3

= x2 + 2x + 3x + 2.3

= x(x+2) + 3(x+2)

= (x+2) (x+3)

Factorization Formula for a Quadratic Polynomial

A “quadratic” is a polynomial that is written like “ax2 + bx + c”, where “a”, “b”, and “c” are just numbers. For an easy case of factoring, you can identify the two numbers that will not only multiply to equal the constant term “c” but also add up to equal “b,” the coefficient on the x-term.

Factorising formulas algebra is especially important when solving quadratic polynomial When reducing formulas we normally have to remove all the brackets, but in particular cases, for example with fractional formulas, sometimes we can use factorisation to shorten a formula.

Solved Examples

Here are some maths factorisation example questions and how to factorise the quadratic polynomial are explained in detail.

x2 + 7x + 6

The constant term is 6, which can be written as the product of 2 and 3 or of 1 and 6. But 2 + 3 = 5, so 2 and 3 are not the numbers I need in this case. On the other hand, 1 + 6 = 7, so you can use 1 and 6:

x2 + 7x + 6 = (x+1)(x+6)

Note that the order doesn’t matter in multiplication, so the above answer can be written as “(x + 6)(x + 1)”.

Q.2: Factorise: x2 – 64.

Solution: Given, x2 – 64

We can also write the given expression as:

⇒ x2 – 82   [Since 8 x 8 = 64]

Now by using the formula,

⇒ a2 – b2 = (a + b) (a – b)

⇒ x2 – 82  = (x + 8) (x – 8)

Q.3: Find the greatest common factor of 6x7 + 3x4 − 9x3.

Solution: Given, 6x7 + 3x4 − 9x3

Now factoring the given expression, we get;

⇒ 3x3 ( 2x4 + x − 3 )

Therefore, the greatest common factor is 3x3.

Q.4: Factorise (7 x + 7 x3) + (x4 + x6).

Solution: (7 x + 7 x3) + (x4 + x6)

Open all the brackets.

⇒ 7x + 7x3 + x4 + x6

Taking the common factors.

⇒ 7x (1 + x2) + x4 (1 + x2)

⇒ (7x + x4) (1 + x2)

⇒ x (7 + x3 ) (1 + x2)

Hence, the required factors.

Practice Questions

1. Factorise 10x2 + x – 2.                                        [Answer: (5x – 2)(2x + 1)]
2. Factorise 4x2 + 12x + 5.                                     [Answer: (2x + 5)(2x + 1)]
3. Factor the polynomial x2 – 10x + 21                [Answer: (x – 3) (x – 7)]

What is Factorisation?

Factorisation is a method of breaking the arithmetic algebraic expressions into the product of their factors. If we multiply the factors again, then they will result in original expression.

How to factorise the given algebraic expression?

There are a few methods or tricks to remember. Reduce the expression by the greatest common factor. Check if there are perfect squares or differences of squares. Use the algebraic formulas or identities to factorise the expression.

The most common method to solve quadratic equations is complete the square method or directly use the quadratic formula.

What is the example of factorisation?

Suppose, 3x+6xy+30+60y is an expression.
Taking the common factors we get;
3x(1+2y)+ 30 (1+2y)
(1+2y) (3x+30)
3(1+2y) (x+10)
Thus, the required factors are:
3(1+2y)(x+10)

What is the rule for factorisation?

If there are four terms in a given expression, then take out the common terms. Suppose, xy-5y+10-2x is given to us. So we can write,
y(x-5) + 2(5-x)
Or
y(x-5)-2(x-5)
Now writing them as product of factors, we get;
(x-5) (y-2)