The factor which is present in each term is taken as common and each term of the expression divided by the common factor and the quotient represents the factorization of the expression. For example,
= axy2 + ax2y
= 2xy(y + 2x)
Factorisation of expressions having a common factor in the group of terms
In this case, the terms of the given expression are arranged in groups in such a way that each group has a common factor. For example,
a2 + ab + ca + bc
a(a+b) + c(a+b)
Factorisation of expressions having the difference of two squared terms
The algebraic expressiona2 - b2 is the difference of two squared terms. Here a2 and b2 are the squared terms and a and b are their square roots respectively. We have learnt that a2 - b2 is the product of (a+b) and(a-b)
\(\therefore\) a2 - b2 = (a+b) (a-b)
So, (a+b) and (a-b) are the factors of a2 - b2
To factorise such expression, we should re-write given terms in the form of a2 - b2.
Then, a2 - b2 = (a+b) (a-b) represents the factorise. For example,
a) 4x2 + 9y2
= (2x)2 - (3y)2
b) 9x2 + 25y2
= (3x)2 + (5y)2
Factorisation of expressions of the form x2 + px + q
While factorising a trimonial expression of the formx2 + px + q, we should search any two number a and b such that a+b = p and ab = q. Clearly a and b must be the factors of q. Then px is expanded in the form ax + bx and factorization is performed by grouping.
a2 + ab + ca + bc
= a(a+b) + c(a+b)
= (a+b) (a+c)
x2 - 3a + 3x - ax
= x2+ 3x - ax - 3a
= x(x+3) -a (x+3)
= (x+3) (x-a)
= (9x2)2 - (4y2)2
= (9x2 + 4y2) (9x2 - 4y2)
= (9x2 + 4y2) [(3x2)2 - (2y)2]
= (9x2 + 4y2) (3x + 2y) (3x - 2y)
Here, (a+b) = 2
\(\therefore\) (a+b)3 = 23
or, a3 + 3a2b + 3ab2 + b3 = 8
or, a3 + b3 + 3ab(a+b) = 8
or, a3 + b3 + 3ab×2 = 8
or, a3 + b3 + 6ab = 8
So, the reqiured value of a3 + b3 + 6ab is 8.
= 24(40 - 20)
= (65+55) (65-55)
What will be factors of 2 and 3?
What will be results of 24 × 40 - 20 × 24 by simplifying?
What will be answered when simplifying 652 - 552
Factorise x2 + 5x + 6
Factorise x2 - 4.
Factorise x2 - 3a + 3x - ax
Resolve into factors 3a(x + y) - 4b(x + y)
Simplify 79 × 81 - 5
Factorise: max - may
Factorise 36 × 24 + 26 × 36