6.2.2 Algebraic Expressions (III), PT3 Practice


6.2.2 Algebraic Expressions (III), PT3 Practice
 
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 = (2+ 3f ) – 3d (2e + 3f)
= (2+ 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


Question 9:
  (a) Factorise 12x2 – 27y2
  (b)   Simplify 3 m 2 10 m + 3 m 2 9 ÷ 3 m 1 m + 3 .  

Solution:
(a)
12x– 27y2 = 3 (4x2 – 9y2)
= 3(2x – 3y) (2x + 3y)

(b)
3 m 2 10 m + 3 m 2 9 ÷ 3 m 1 m + 3 = ( 3 m 1 ) ( m 3 ) ( m + 3 ) ( m 3 ) × m + 3 3 m 1 = 1
  

Question 10:
Simplify:  8m+mn 3m ÷ n 2 64 24

Solution:
8m+mn 3m ÷ n 2 64 24 = 8m+mn 3m × 24 n 2 64 = m( 8+n ) 3m × 24 n 2 8 2 = m ( 8+n ) 3 m × 24 8 ( n8 ) ( n+8 ) = 8 n8

6.2.1 Algebraic Expressions (III), PT3 Practice


6.2.1 Algebraic Expressions (III), PT3 Practice

Question 1:
(a)(i) Factorise 18a + 3
(a)(ii) Expand –3 (–y + 5)
(b) Express 5 6 y 3 x 5 12 y as a single fraction in its simplest form.

Solution:
(a)(i) 18+ 3 = 3(6a + 1)
 
(a)(ii) –3 (–+ 5) = 3y – 15
 
(b)
5 6 y 3 x 5 12 y = 5 × 2 6 y × 2 ( 3 x 5 ) 12 y = 10 3 x + 5 12 y = 15 3 x 12 y = 3 ( 5 x ) 4 12 y = 5 x 4 y


Question 2:
(a) Expand:
 (i) 3 (–a + c)
 (ii) –5 (c)
(b) Factorise 4+ 2
(c) Simplify: 3 x + 6 x 2 4 ÷ x + 2 x 2   

Solution:
(a)(i) 3 (–+ c) = –3a + 3c
 
(a)(ii) –5 (c) = –5a + 5c
 
(c)
3 x + 6 x 2 4 ÷ x + 2 x 2 = 3 ( x + 2 ) ( x + 2 ) ( x 2 ) × x 2 x + 2 = 3 x + 2



Question 3:
(a) Factorise:
 (i) 5m + 25
 (ii) 7x + 9xy
(b) Simplify: 4 x 12 4 y ÷ x 2 9 y z

Solution:
(a)(i)5m + 25 = 5 (m + 5)
 
(a)(ii)7x + 9xy = x (7 + 9y)

(b) 4 x 12 4 y ÷ x 2 9 y z = 4 ( x 3 ) 4 y × y z ( x + 3 ) ( x 3 ) = z x + 3


Question 4:
  (a) Factorise completely:
  4 – 100n2
(b) Express 4 5 x 7 10 y 15 x  as a single fraction in its simplest form.

Solution:
(a)
4 – 100n2= (2 + 10n)(2 – 10n)
(b)
4 5 x 7 10 y 15 x = 4 × 3 5 x × 3 ( 7 10 y ) 15 x = 12 7 + 10 y 15 x = 5 + 10 y 15 x = 5 ( 1 + 2 y ) 3 15 x = 1 + 2 y 3 x


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)(+ 4n) – m2
= m2 + 4mn – 4mn – 4n2m2
= 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


6.1 Algebraic Expressions III


6.1 Algebraic Expressions III

6.1.1 Expansion
1. The product of an algebraic term and an algebraic expression:
  • a(b + c) = ab + ac
  •  a(bc) = ab ac
2. The product of an algebraic expression and another algebraic expression:
  • (a + b) (c + d)  = ac + ad + bc + bd
  • (a + b)2= a2 + 2ab + b2
  • (ab)2= a2 – 2ab + b2
  • (a + b) (ab) = a2b2

6.1.2 Factorization
1. Factorize algebraic expressions:
  •  ab + ac = a(b + c)
  • a2b2 = (a + b) (ab)
  • a2+ 2ab + b2 = (a + b)2
  • ac + ad + bc + bd = (a + b) (c + d)  
2. Algebraic fractions are fractions where both the numerator and the denominator or either the numerator or the denominator are algebraic terms or algebraic expressions.
Example:
3 b , a 7 , a + b a , b a b , a b c + d


3(a) Simplification of algebraic fractions by using common factors:
1 4 b c 3 12 b d = c 3 d b m + b n e m + e n = b ( m + n ) e ( m + n ) = b e

3(b) Simplification of algebraic fractions by using difference of two squares:
a 2 b 2 a n + b n = ( a + b ) ( a b ) n ( a + b ) = a b n
 

6.1.3 Addition and Subtraction of Algebraic Fractions
1. If they have a common denominator:
a m + b m = a + b m

2.
If they do not have a common denominator:
a m + b n = a n + b m n m


6.1.4 Multiplication and Division of Algebraic Fractions
1. Without simplification:
a m × b n = a b m n a m ÷ b n = a m × n b = a n b m  

2. 
With simplification:
a c m × b m d = a b c d a c m ÷ b d m = a c m × d m b = a d b c