FACTORISATION
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Factors: If an algebraic expression is written as the product of numbers or algebraic
expressions, then each of these numbers and expressions are called the factors of the given
algebraic expression and the algebraic expression is called the product of these expressions.
Factorization: The process of writing a given algebraic expression as the product of two or
more factors is called factorization.
Greatest Common Factor (GCF) Or Highest Common Factor (HCF)
The greatest common factor of given monomials is the common factor having greatest
coefficient and highest power of the variables.
The following step-wise procedure will be helpful to find the GCF of two or more
monomials.
Step I Obtain the given monomials.
Step II Find the numerical coefficient of each monomial and their greatest common factor.
Step III Find the common literals appearing in the given monomials.
Step IV Find smallest power of each common literal.
Step V Write a monomial of common literals with smallest powers obtained in step IV.
Step VI The required GCF is the product of the coefficient obtained in step II and the
monomial obtained in step V.
Factorization of Algebraic Expression when a common Monomial Factor Occurs in each Term
In order to factories algebraic expressions consisting of a common monomial factor of each
term we use the following step-wise procedure
Step I Obtain the algebraic expression.
Step II Find the greatest common factor (GCF/HCF) of its terms.
Step III Express each term of the given expression as the product of the GCF and the quotient
when it is divided by the GCF.Step IV Use the distributive property of multiplication over addition to express the given
algebraic expression as the product of the GCF and the quotient of the given expression by the GCF.Factorization of Q
Excersice 12.1
Question 1.
Find the common factors of the given terms in each.
(i) 8x, 24
(ii) 3a, 2lab
(iii) 7xy, 35x2y3
(iv) 4m2, 6m2, 8m3
(v) 15p, 20qr, 25rp
(vi) 4x2, 6xy, 8y2x
(vii) 12 x2y, 18xy2
Solution:
8x = 2 × 2 × 2 × x
24 = 8 × 3 = 2 × 2 × 2 × 3
∴ Common factors of 8x, 24 = 2, 4, 8.
ii) 3a, 2lab
3a = 3 × a
21ab = 7 × 3 × a × b
∴ Common factors of 3a, 21ab = 3, a, 3a.
iii) 7xy, 35x2y3
7xy = 7 × x × y
35x2y3 = 7 × 5 × x × x × y × y × y
∴ Common factors of 7xy, 35x2y3
= 7, x, y, 7x, 7y, xy, 7xy
iv) 4m2, 6m2, 8m3
4m2 = 2 × 2 × m × m
6m2 = 2 × 3 × m × m
8m3 = 2 × 2 × 2 × m × m × m
∴ Common factors of 4m2 , 6m2 , 8m3
= 2, m, m2, 2m, 2m2.
v) 15p, 20qr, 25rp
15p = 3 × 5 × p
20qr = 4 × 5 × q × r
25rp = 5 × 5 × r × p
∴ Common factors of 15p, 20qr, 25rp = 5.
vi) 4x2, 6xy, 8y2x
4x2 = 2 × 2 × x × x
6xy = 2 × 3 × x × y
8y2x = 2 × 2 × 2 × y × y × x
∴ Common factors of 4x2, 6xy, 8xy2 = 2, x, 2x.
vii) 12x2y, 18xy2
12x22y = 2 × 2 × 3 × x × x × y
18xy2 = 3 × 3 × 2 × x × y × y
∴ Common factors of 12x2y, 18xy2
= 2,3, x, y, 6, xy, 6x, 6y, 2x, 2y, 3x, 3y, 6xy.
Question 2.
Factorise the following expressions
(i) 5x2 – 25xy
(ii) 9a2 – 6ax
(iii) 7p2 + 49pq
(iv) 36 a2b – 60 a2bc
(v) 3a2bc + 6ab2c + 9abc2
(vi) 4p2 + 5pq – 6pq2
(vii) ut + at2
Solution:
(i) 5x2 – 25xy
= 5 x × x × – 5 × 5 × x × y
= 5 × x [x – 5 × y]
= 5x [x – 5y]
ii) 9a2 – 6ax
= 3 × 3 × a × a – 2 × 3 × a × x
= 3a [3a – 2x]
iii) 7p2 + 49pq
= 7 × p × p +7 × 7 × p × q
= 7p[p + 7q]
iv) 36a2b – 60a2bc
= 2 × 2 × 3 × 3 × a × a × b – 2 × 2 × 3 × 5 × a × a × b × c
= 2 × 2 × 3 × a × a × b[3 – 5c]
= 12a2b [3 – 5c]
v) 3a2bc + 6ab2c + 9abc2
= 3 × a × a × b × c + 3 × 2 × a × b × b × c + 3 × 3 × a × b × c × c
= 3abc [a + 2b + 3c]
vi) 4p2 + 5pq – 6pq2
= 2 × 2 × p × p + 5 × p × q – 2 × 3 × p × q × q
= p [4p + 5q – 6q2]
vii) ut + at2
= u × t + a × t × t = t [u + at]
Question 3.
Factorise the following:
(i) 3ax – 6xy + 8by – 4bx
(ii) x3 + 2x2 + 5x + 10
(iii) m2 – mn + 4m – 4n
(iv) a3 – a2b2 – ab + b3
(v) p2q – pr2 – pq + r2
Solution:
i) 3ax – 6xy + 8by – 4ab
= (3ax – 6xy) – (4ab – 8by)
= (3 × a × x – 2 × 3 × x × y)
– (4 ×a × b – 4 × 2 × b × y)
= 3x(a – 2y) – 4b(a – 2y)
= (a – 2y)(3x – 4b)
ii) x3 + 2x2 + 5x + 10
= (x3 + 2x2) + (5x +10)
= (x2 × x + 2 × x2) + (5 × x + 5 × 2)
= x2(x + 2) + 5(x + 2)
= (x + 2) (x2 + 5)
iii) m2 – mn + 4m – 4n
= (m2 – mn) + (4m – 4n)
= (m × m – m × n) + (4 × m – 4 × n)
= m(m – n) + 4(m – n)
= (m – n) (m + 4)
iv) a3 – a2b2 – ab + b3
= (a3 – a2b2) – (ab – b3)
= (a2 × a – a2 × b2) – (a × b – b × b2)
= a2(a – b2) – b(a – b2)
= (a – b2) (a2 – b)
v) p21 – pr2 – pq + r2
= (p2q – pr2) – (pq – r2)
= (p × p × q – p × r × r) – (pq – r2)
= p(pq – r2) – (pq – r2) × 1
= (p – 1) (pq – r2)
Excersice 12.2
