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2
2
2
Alternatively, 3m – 12m + 12 = 3(m – 4m + 4) = 3(m – 2m – 2m + 4)
= 3[m(m – 2) – 2(m – 2)] = 3(m – 2)(m – 2)
2
2
(b) 10x + x – 3 = 10x + 6x – 5x – 3 [Q 1 × –3 = –30 and 6 + (–5) = –1]
2
= 10x – 5x + 6x – 3
= 5x(2x – 1) + 3(2x – 1) = (5x + 3)(2x – 1)
Practice Time 13B

1. Factorise the following by using suitable identities.
2
2
2
2
(a) x + 6x + 9 (b) x – 18x + 81 (c) 9p – 12p + 4 (d) 49m + 84mn + 36n 2
4x 2
2
2
3
4
2 2
4
2
(e) p – 14pq + 49q 2 (f) x + 2x y + y 4 (g) 9y – 4xy + (h) t – 4t + 4t
9
2. Factorise the following.
x 2 y 2
2
2
(a) 25x – 81y 2 (b) 676 – p 2 (c) − (d) (5a – b) – 16c 2
64 49
1 16
22
4
2
(e) m – (2n – 3) 2 (f) (a + b) – (a – b) 2 (g) ab − bc (h) z – 625
2
22
36 49
3. Factorise by splitting the middle terms.
2
2
2
2
(a) x + 9x + 20 (b) m – 11m – 102 (c) y + 4y – 21 (d) 12x – 29x + 15
2
2
2
2
(e) x + 10xy + 21y 2 (f) 7x + 35x + 42 (g) 4m – 8m + 3 (h) 12p + 7p – 10
Division of Algebraic Expressions Using Factorisation
Division of algebraic expressions means dividing one expression by another. These expressions
include variables, constants, and operations like addition, subtraction, multiplication, and division.
There are different ways to divide algebraic expressions, such as:
• Dividing a monomial by a monomial
• Dividing a polynomial by a monomial
• Dividing a polynomial by a polynomial
Dividing a Monomial by a Monomial
Remember
This is the simplest case where both the dividend and the
divisor are monomials. In this case, both monomials are When the dividend and divisor have the
factorised and the common factors are cancelled out. same signs, the quotient has a plus sign.
When the dividend and divisor have
Example 10: Divide opposite signs, the quotient has the
3 2
4
(a) 63x by 7x 3 (b) 51x y z by 17xyz negative sign.
Solution: (a) We have,
63x 4 79x 4 7 ×× x 3 × x
×
9
3
4
63x by 7x = = = = 9x
7x 3 7x 3 7 × x 3
(b) We have,
x
y
x
3
51xy z 17 3xy z 17 ×× x ××× y ×× z
2
2
3
3
×
2
51x y z by 17xyz = = = = 3x y
3 2
17xyz 17xyz 17 × x × y × z
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