2.2.2 Factorisation and Algebraic Fractions, PT3 Practice


Question 5:
(a) Simplify:
(m – 4n)(m + 4n) – m2
(b) Simplify: 3 x − 3 y x + y × 2 x + 2 y 6 x  

Solution:
(a)
(m – 4n)(m + 4n) – m2
= m2 + 4mn – 4mn – 4n2 – m2
= 0
 
(b)
3 x − 3 y x + y × 2 x + 2 y 6 x = 3 ( x − y ) x + y × 2 ( x + y ) 6 x = x − y x



Question 6:
(a) Simplify each of the following:
(i) 12 m n 32 (ii) x 2 − x y x
(b) Express 1 2 q − 2 p − 7 6 q  as a single fraction in its simplest form.

Solution:
(a)(i) 12 m n 32 = 3 m n 8 (a)(ii) x 2 − x y x = x ( x − y ) x = x − y

(b)
1 2 q − 2 p − 7 6 q = 1 × 3 2 q × 3 − ( 2 p − 7 ) 6 q = 3 − 2 p + 7 6 q = 10 − 2 p 6 q = 2 ( 5 − p ) 3 6 q = 5 − p 3 q


Question 7:
(a) Factorise 2ae + 3af – 6de – 9df
(b)   Simplify a 2 − b 2 ( a + b ) 2

Solution:
(a)
2ae + 3af – 6de – 9df = a (2e + 3f ) – 3d (2e + 3f)
= (2e + 3f ) (a – 3d)

(b)
a 2 − b 2 ( a + b ) 2 = ( a + b ) ( a − b ) ( a + b ) ( a + b ) = a − b a + b


Question 8:
(a) Factorise –8c2 – 12ac.
(b)   Simplify a e + a d − 2 b e − 2 b d a 2 − 4 b 2 .  

Solution:
(a)
–8c2– 12ac
= –4c (2c + 3a)

(b)
a e + a d − 2 b e − 2 b d a 2 − 4 b 2 = a ( e + d ) − 2 b ( e + d ) ( a + 2 b ) ( a − 2 b ) = ( e + d ) ( a − 2 b ) ( a + 2 b ) ( a − 2 b ) = e + d a + 2 b

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