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Factorization is basically the process of writing any algebraic expression as the product of two or more factors. These factors can be numbers, algebraic variables and algebraic expressions. The formulas that we used to solve algebraic expressions are called factoring formulas. If we write any algebraic expression as the product of numbers or algebraic expressions, then each of these numbers and expressions are called the factors of the given algebraic expressions and the algebraic expressions are termed as the product of expressions.
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Key Terms: Factorization, Binomial, Monomial, Formulas of Factorization, Algebraic Variables
Also read: Isosceles Triangle Theorems
Factorization
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- Factorization can be done in various ways and using multiple factoring formulas. The methods of factorization and factoring formulas are mentioned here:
- Factorization when a common monomial factor occurs in each term.
- Factorization when a binomial is common.
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Factoring Formulas
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Factorization can also be done by using direct formula, few factoring formulas are given here:
Formula of factorization when a given expression is a difference of two squares
(a2-b2) = (a+b)(a-b)
Example: (a). 48a2 – 243 b2 (2 Marks)
Comparing the equation with (a2- b2) = (a + b) (a-b)
If we divide the equation with 3
= 3(16 a2 – 81b2)
= 3[ (4a)2 – (9b)2]
= 3 ( 4 a + 9 b) (4 a – 9 b)
(b). - 2 – 50 x2
= 2 {1 – 25 x 2}
= 2 {1 2 – (5 x )2}
Comparing the equation with formula (a2- b2) = (a + b) (a-b)
= 2 ( 1 – 5 x ) (1 + 5 x)
Factoring formula when a given expression is a perfect square
a2 + b2 + 2ab = (a + b) 2
a2 + b2 – 2ab = (a – b)2
Factoring formula of Quadratic Trinomials
x 2 + px + q
x2 +px +q = x2 + (a + b) x + ab
= (x +a) (x+ b)
Example: (a)- - 2x2 – 3 x + 2
- 2 x 2 – 3 x + 2 = - 2 x 2 – 4 x + x + 2
= - 2 x (x + 2) + 1 (x + 2)
= (x + 2) (- 2 x + 1)
(b)- 6 x2 + 35 x y – 6 y2
= 6 x 2 + 36 x y – x y – 6 y2
= 6 x (x + 6 y) – y (x – 6 y)
= (6 x – y) (x + 6 y)
Factoring formula when given expression is a perfect cube
(a+ b)3 = a3+ b3 +3 ab (a+ b)
(a3 – b3) = a3 – b3- 3 ab(a-b)
Factoring Formula with three expressions
(a + b + c) 2 = (a2 + b2 +c2 + 2ab + 2bc + 2ca)
Factoring Formula when the expression is perfect cube
X3 + y3 = (x + y) ( x2- x y – y2)
X3 – y3 = (x-y) (x2+ x y +y2)
Things to Remember
- Factorisation is the reverse process of multiplication.
- Algebraic equations consist of just one term are called monomials and algebraic expressions that are the sum of exactly two monomials are called binomial and trinomial is the sum of three monomials.
- If the expression is the difference of two squares. To factorise it we use the following formula: (a2- b2) = (a+ b)(a-b)
- If the expression is complete square the formula used for factorization is:
a2 + b2 + 2ab = (a + b)2
a2 + b2 – 2 ab = (a – b)2
- x2 + x(a+ b) + ab type of expressions can be solved in the form (x+ a) (x+ b).
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Sample Questions
Ques. Factorise the following algebraic expressions: 3x + 15. (2 Marks)
Ans. The greatest common factor of the two terms 3x and 15 of the expression 3x+15 is the 3.
3x = 3 X x and that of 15 = 3 x 5
Hence: 3x + 15 = 3(x + 5)
Ques. (b) 3 x2 y – 6 x y2
The greatest common factor of the two terms 3 x2 y and 6 x y2 of the binomial 3x 2y – 6 x y2 is 3 x y.
So, 3 x2 y = 3 x y X x and 6 xy2 = 3 x y X 2 y
3x2y – 6 xy2
3 x y (x – 2y)
Ques. Factorise 7(2x+5)+3(2x+5). (2 Marks)
Ans. Taking (2?+5) common
(7(2x+5)+3(2x+5)
(2x+5)(7+3)
10(2x+5) )
Ques. Factorise 9 a2 – 16 b2 (2 Marks)
Ans. (a2 - b2) = (a+ b) (a-b)
9 a2 = (3 a)2
16 b2 = (4 b )2
(3a)2 = a, (4 b)2 = b
A2 – b2 = ( 3 a + 4 b) ( 3 a – 4 b)
Ques. Factorise 3 a4 – 48 b4 (2 Marks)
Ans. As we know the formula: (a2- b2) = (a+ b) (a-b)
= 3(a4- 16 b4)
= 3 (a2)2 – (4 b2)2
=3( a2 – 4 b2) ( a2 + 4 b2)
= 3 ( a2 –(2b)2)( a2 + 4 b2)
= 3(a + 2 b) (a – 2b) (a2 + 4 b2)
Ques. Factorise 1- 2 ab – (a2 + b2). (2 Marks)
Ans. Comparing the given equation to the perfect square formula
(a2 + b2) = a2 + b2 + 2 ab
= 1 – (2 a b + a2 + b2)
= 1 – (a + b) 2
= {1 + (a + b)} {1 - (a + b)}
= (1 + a + b) (1 – a + b)
Ques. Factorise: x2 + 8 x + 15 (2 Marks)
Ans. = X2 + 3 x + 5 x + 15
= (x2 + 3 x) ( 5 x + 15)
= x (x+ 3) 5 (x + 3)
= (x + 5) ( x + 3)
Ques. Factorise: 9 – a6 + 2a3b3 – b6 (2 Marks)
Ans. 9 – a6 + 2 a3b3 – b 6
= (a6 – 2 a3b3 + b6)
= 32 – [(a3)2 – 2 x a3x b3 +(b3)2]
= 32- (a3 – b3)2
= (3 – (a3- b3)) (3 + ( a3 – b3))
= (a3 – b3+ 3) (-a3 + b3 +3)
Ques. Factorise: ( 2x + 3 y)2 - 5( 2x + 3y ) -14. (2 Marks)
Ans. Let say 2 x + 3 y is equal to b
Then the equation will be written as
= b2 – 5 b -14
= b2 – 7 a + 2 a – 14
= b(b-7) + 2(b- 7)
= (b + 2) (b – 7)
Or
= (2 x + 3 y – 7) (2 x + 3 y + 2)
Ques. Factorise:3 m 2 + 24 m + 36 (2 Marks)
Ans. Taking 3 common:
= 3 (m2 + 8 m + 12)
= 3( m2 + 6 m + 2 m + 12 )
= 3 (( m2 + 6 m) + (2 m + 12) )
= 3 ( m( m + 6 ) + 2 ( m + 6 ) )
= 3 ( m + 6 ) ( m + 2 )
Ques. Factorise: x 4 + 4 (2 Marks)
Ans. = x 4 + 4 x 2 + 4 – 4 x 2
= { ( x 2 )2 + 2 X x2 X 2 + 2 2 } – 4 x2
According to formula ( a + b ) 2 = a 2 + b 2 + 2 ab
= ( x + 2 )2 - 2 x 2
= {( x2 + 2) + 2 x2 } { ( x2 + 2) - 2 x 2 }
On simplifying:
= ( x2 + 2 x + 2 ) ( x2 – 2 x + 2)
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