Chapter 7 Algebraaic Expressions

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CHAPTER 7 : ALGEBRAIC EXPRESSIONS

A. Unknown *An unknown is a quantity whose value has not been determined. *Letters can be used to represent unknowns or objects. Example Exercise 1. The teacher gives some pencils 1. I bought some books to the students Solution : …………………………… Solution : The unknown is the number of pencils 2. There are many monkeys in the garden. Solution : ……………………………. 2. There are x students in m class Solution : x is unknown 3. Azman bought y durian in z shop m is object yesterday. Solution : unknown…………….. object ………………. 4. Mr a sold his car for k ringgit Solution : unknown ……………….. object :………………….. B Algebraic Terms i) Algebraic Term with one unknown - is the product of an unknown and a number. Example : 4y is called an algebraic term 4y Number

4y = 4 x y = y + y + y + y

unknown

* Identify coefficients in given algebraic term - Coefficient is the number that multiply the unknown Example 1) 7m : coefficient of m is 7 2)

r 1 : coefficient of r is 4 4

3) – y : coefficient of y is -1

Exercise 1) -3z : coefficient of z is…………… 2)

2 x : coefficient of x is ………….. 5

3) 0.7 h : coefficient of h is ………… 4) p : coefficient of p is …………..

Algebraic Expressions

77

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(ii) Like and Unlike Algebraic Terms * Like term : terms with the same unknowns * Unlike terms : terms with different unknowns. Example 1. 3m and -4m 2. 4x and ¼ x 3. 0.9z and 5z 1. 2w and 8h 2, -5f and ½g 3. 1.2q and 3.5g

Exercise Determine whether each of the Like term following pairs of algebraic terms are (same unknown) like term / unlike term 1. 6s , - t : …………………… y 2. , 8y : …………………….. 4 3. 19 d , 19e : …………………… 2 4. , 4e : ……………………….. e

unlike term (different unknown)

C Algebraic Expressions An algebraic expression is a combination of two or more algebraic terms by addition, subtraction or both Examples : 2x + 4y , 6r – 3s + 6z (i)

Number of terms in a given algebraic expression

Example Determine the number of terms in the algebraic expressions below :

Exercise Determine the number of terms in the algebraic expressions below :

1. 3x + 6y : 2 terms

1. 6m + 8n – 9 : …………………..

2. 7p + 5q – 9 : 3 terms

2. 3b + 2e – 10b -5e : …………….

3. w – 2z – 8y + 1 : 4 terms

3. 2s – 4s + 5s + 3 – w :………….

(ii) Simplifying Algebraic Expressions - Group all the like terms together - Add / subtract the coefficient of the terms - unlike terms cannot be simplified

Algebraic Expressions

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Examples 1. 6m – 2n + 4m – 5n = 6m + 4m -2n -5n (group like terms) = 10m – 7n

Exercise .1. 2x – 7y + 5x – y =

2. -7x + 4y + 3y + 2x = -7x + 2x + 4y + 3y = -5x + 7y

2. 11z -3w - 8z- 8w =

3. ( 12a – 4b) + ( 5a + 7b) = 12a+ 5a – 4b + 7b = 17a + 3b

3. ( 6r + 9s) + ( 3r – 2s) =

4. ( 9q + 2p) – ( 4q – 6p) = 9q + 2p – 4q + 6p = 9q – 4q + 2p + 6p = 5q + 8p

4. ( 5k -3) – ( 7k + 2) =

5. 8x – ( - 4x) + x = 8x + 4x + x = 13x

5. (2t +4s) – (7t – 3s) =

6. -3c –(- d) +(-2d) = -3c +d – 2d =-3d -d

6. – 3s – (- 5s + 1) =

7. -14w –(-3w) -7w =

D. Algebraic Terms in two or more unknowns Is the multiplying factors of the term Examples : 3xy , ½abc, 0.8 def * Identifying the coefficient of an unknown Example In the term 8xy2 * 8y(xy) the coefficient of xy is 8y * 8x(y2) the coefficient of y2 is 8x * 8y2 (x) the coefficient of x is 8y2 * 8(xy2) the coefficient of xy2 is 8 Algebraic Expressions

Exercise 1. in the term -3ab2c * the coefficient of abc =……………….. * the coefficient of ab2 = ……………….. * the coefficient of ab2c =…………….. * the coefficient of ac = ………………. 79

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E. Multiplication & Division of 2 or more terms (i) Finding the product of 2 algebraic terms - collect all numbers and similar unknowns together - then multiply the numbers and the unknown separately. Example 1. 2ab x 4b2c =2xaxbx4xbxbxc =2x4xaxbxbxbxc = 8 x a x b2 x c = 8ab3c

Exercise 1. ab x a2b =

2. 3xy x (-2 yz) 2

2

2. 4m x ½ mn =4x½xmxmxmxnxn = 2 x m 3 x n2 = 2 m3 n2 Exercise 4 1 pqr × (− pqr ) 6. 5 2 =

7. (-3m2hk3)x (-7m2hk2) =

=

3.. 6ab2c x (½ bc3) =

4. (-8p3qr) x ( -7pqr2) =

5. − =

Algebraic Expressions

80

2 2 w z × ( −9 wz 3 ) 3

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(ii) Finding the quotient of two algebraic terms - Express the division in fraction form - cancel similar unknowns that are found in both numerator and denominator Example 14 xyz 1. 7 xz 14 2 × x × y × z = 7× x× z = 2y 2. 12m2n ÷ 3mn 12 4 × m × m × n = 3× m × n = 4m 3. -5cd2e ÷ 15c2de (−5) × c × d × d × e = 3 15 × c × c × d × e d = − 3c

Exercise 1. 24pq2z ÷ 8qr =

− 16 x 3 y 2 2. − 4x 2 y 3 =

3. 12abc ÷ (-18cd) =

4. (- 18sr3t2) ÷ 6sr2t =

iii) Multiplication and Division involving algebraic terms Example 1. 4p x 6q2 ÷ 3pq 4 × p × 62 × q × q = 3× p × q = 8q − 4c 2 de × 9cde 2 6ce 2 − 4 2 × c × c × d × e × 93 × c × d × e × e = 6 31 ×c × e × e

2.

= -2 x c x d x 3 x c x d x e = -2 x 3 x c x c x d x d x e = -6c2d2e Algebraic Expressions

Exercise 1. 6p2qr ÷ 3pq x 8pr =

2.

− 12 gh × (−9kh) 6 gh 2 =

3. 10a2b3 x (-2b2c) ÷ 5abc 81

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F) Computations involving Algebraic Expressions * In Multiplication / division of algebraic expressions by a number, every term In the expression is multiply/ divide by the same number Example Exercise 1. 3 ( 2a –b) 1. 8 ( 5m -2) = 3 x 2a – 3 x b = = 6a – 3b 2

1 − (5t + 10 s ) 5 1 1 = − × 5t − × 10 2 s 5 5 = −t − 2 s

3. h – 9(h – 2) = h – 9h + 18 = -8h +18

4. 2 (4e +y) – 5( 2e – 3y) = 2 x 4e + 2 x y – 5 x 2e + 5 x 3y = 8e + 2y – 10e + 15y = 8e -10e +2y + 15y = -2e + 17y 5. (6ab – 4bc) ÷ 2b = 6ab ÷ 2b – 4bc ÷ 2b = 3a – 2c 6.

2. - ½ ( 4a + 12b) = 3. – p – 7 ( p -3 ) =

4. – 5 ( t -2) + 8t = 5. 3 ( 2s -7) – 4( s + 3) =

6. . ( -12pq + 8qr – 4pqr) ÷ 4 =

7.

1 3 (4 p 2 − 8 pq) − (4 p 2 − 12 pq ) 2 4 2 = 2 p − 4 pq − 3 p 2 + 9 pq = 2 p 2 − 3 p 2 − 4 pq + 9 pq = − p 2 + 5 pq 12 x − 4 4 = 10x -5 – (3x - 1) = 10x -5 -3x + 1 = 10x -3x -5 + 1

8.

3x − 9 − 5y 3 =

1 2 (5 p 2 − 10 pq) − (6 p 2 − 3 pq) 5 3 =

7. 5(2x – 1) -

Algebraic Expressions

9. 82

18 − 24u 2 + 5(2u 2 − 3) 6

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= 7x -4

=

Common Errors Errors 1. 7pq x 3pq = 21 pq

Correct Steps 1. 7pq x 3pq =7x3xpxpxqxq = 21 p2q2

2. 2 ( 4e – 3 d) = 8e – 3d

2. 2( 4e – 3d) = 2 x 4e – 2 x 3d = 8e – 6d

3. (6de2 – 4ef) ÷ 2e = 3de – 4ef

3. (6de2 – 4ef) ÷ 2e = 6de2 ÷ 2e – 4ef ÷ 2e = 3de – 2f

4. (x – 4y) – ( 2x + y) = x – 4y – 2x + y = x -2x -4y + y = -x-3y

4. (x – 4y) – ( 2x + y) = x – 4y – 2x - y = x -2y -4y - y = - x - 5y

5. -2p ( pq – 3) = - 2pq – 6p

5. -2p ( pq – 3) = - 2p2q + 6p

6. 10abc – 4 abc =6

6. 10abc – 4 abc = 6abc

7. 3a +6b – 8a – 3b = 3a + 8a - 6b -3b = 11a – 9b

7. 3a +6b – 8a – 3b = 3a - 8a + 6b -3b = -5a +3b

8. ( - 4rs2t) x 5r3st2 = (-4) x 5 x r x r3 x s2 x s x t x t2 = 20r3s2t2

8. ( - 4rs2t) x 5r3st2 = (-4) x 5 x r x r3 x s2 x s x t x t2 = -20r4s3t3

9. -5s - ( 3t – 2)

9. -5s - ( 3t – 2) = -5s -3t +2

= +15st -10s Algebraic Expressions

83

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(G) Expanding single Brackets * Expanding algebraic expressions by multiplying each term inside the bracket by the number or term outside * p(q + r)= p × q + p × r = pq + pr Example 1. 2p (p – 3q) = 2p x p – 2p x 3q = 2p2 – 6pq 2. -4b(2a + b) = -4b x 2a -4b x b = -8ab – 4b2 3.

2 b (6a – 9c) 3 3 2 2 2 = b x 6 a - b x 9c 3 3 = 4ab -6bc

Exercise 1. y ( w + y) =

2. -5e ( 3f + 2g) =

3.

r (12 − 9 s + 3t ) 3 =

4. xy ( 4z – 2w + xy) =

5.

2 x( xy − 5 yz 2 + 10 z ) 5 =

6. - 7ab(2a – 4b + c) = Algebraic Expressions

84

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(H) Expanding double brackets * Expanding algebraic Expressions by multiplying each term within the first pair of brackets by every term within the second pair of brackets ( a + b)(x + y) = a( x +y) + b( x+y) = ax + ay + bx + by Example 1. (x -3)(y+5) = x (y + 5) – 3(y + 5) = xy + 5x – 3y – 15 2. (2k -1)(k – 3) = 2k(k -3) – 1(k -3) = 2k2- 6k – k + 3 = 2k2 -7k + 3 3. (p – 3q)2 = (p – 3q)(p -3q) = p(p-3q) – 3q (p-3q) = p2 – 3pq – 3pq + 9q2 = p2 – 6pq + 9q2 4. (2a +b)2 = (2a+b)(2a+b) = 2a(2a+b) + b(2a+b) = 4a2 +2ab +2ab + b2 = 4a2 +4ab +b2

Exercise 1. (a -2)(b +1) =

2. (m +3)( 3m – n) =

3. (-2s -5)( 3t + 4) =

4. ( a -3)2 =

5. (3m –n)2 =

6. ( 5x +2)2 =

Algebraic Expressions

85

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7. (y + 4d)2 =

Common Errors Errors 1. 2x(x-3)

Correct Steps 1. 2x(x-3)

= 2x2 -3 2. (a+b)2

= 2x2 – 6 2. (a+b)2

= a2 + b2 3. ( a – b)2

= a2 +2ab + b2 3.. ( a – b)2

= a2 – b2 4

2m2(3m2n – 4 mn3)

= a2 – 2ab + b2 4.

= 6m4n – 4m2n3 5. -4 ( 3de – 2rst2)

= 6m4n – 8m3n3 5. -4 ( 3de – 2rst2)

= - 12de – 2rst2 6. ( x -3)2

= - 12de + 8rst2 6. ( x -3)2

= x2 – 9 7. 4a2 –(a + b)2

2m2(3m2n – 4 mn3)

= x2 – 6x + 9 7. 4a2 –(a + b)2

= 4a 2 –a2 + b2

= 4a 2 –( a2 +2ab + b2 )

= 3a2 + b2

= 4a2 – a2 - 2ab - b2 = 3a2 -2ab –b2

8. (2x -3)(x + 4)

8. (2x -3)(x + 4)

= 2x( x+4) – 3 (x + 4) Algebraic Expressions

= 2x( x+4) – 3 (x + 4) 86

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= 2x2 + 8x – 3x + 12

= 2x2 + 8x – 3x - 12

= 2x2 +5x + 12

= 2x2 +5x – 12

(I ) Factorization * Process of writing an expression as a product of two or more factors. -

List out common factors for each alg. term , determine the HCF of the terms . Write as the product of 2 factors

Example 1. st – sr = s( t – r )

ab – ac = a ( b – c) a = common factor Exercise 1. 6a – 24c =

2. 4m + 12mn – 16m2 2. 4m3 – 6m2 = 4 xm + 4 x 3x mxn- 4 x 4 x m x m = = 4m ( 1 + 3n – 4m) 3. 6d2 – 3d = 3 x 2 x d xd – 3 x d = 3d ( 2d – 1)

3. 8ax + 4bx – 2cx =

4. 10mn – 15m2 =5x2xmxn–5x3xmxm = 5m ( 2n – 3m )

4. x2yz – xy2z =

5. 3st2 – 15 stw =

6. 2yz – 4yz2 + 6xyz = Algebraic Expressions

87

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*Factorize an expression by using the difference between 2 squares i) expressions which consist of 2 terms : a2 – b2 = ( a – b)( a + b) Example 1 9 – a2 = 32 – a2 = ( 3-a)(3+a)

Exercise 1. w2 – 25 =

2. 4x2 – 25y2 = 22 x2 – 52 y2 = ( 2x)2 – (5y)2 = ( 2x – 5y)( 2x + 5y)

2. 5x2 -5 =

3. 8g2 – 18h2 = 2 ( 4g2 – 9h2 ) = 2 [ ( 2g)2 – ( 3h)2 ] = 2 (2g -3h) (2g + 3h)

3. 12d2 – 75 =

4. 36c2 -100e2 =

ii) expressions which consist of 3 terms a2 + 2ab + b2 = (a + b)2 a2 - 2ab + b2 = (a - b)2 Example 1. 9x2 + 6xy + y2 = (3x2) + 2 (3x)(y) + y2 = ( 3x + y)2 2. p2 – 4pq + q2 = p2 – 2(p)(q) + q2 = (p – q)2 Algebraic Expressions

Exercise 1. a2 + 4ab + b2 =

2. 4x2- 20x + 25 = 88

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3. 16p2 – 24pq + 9q2 = (4p)2 – 2(4p)(3q) +(3q)2 = (4p –3q)2

3. 9e2 – 12 ef + 4f2 =

iii) expressions which consist of 4 terms ax + ay - bx - by = (ax + ay) -(bx + by) = a( x+y) - b(x+ y) = (a -b)(x + y)

ax + ay + bx + by = (ax + ay) + (bx + by) = a( x+y) + b(x+y) = (a+b)(x+y) Example 1. w2 + wz + 6w + 6z

Exercise 1. pq + qr + ps + rs

= (w2 + wz) +( 6w + 6z)

=

= w(w+z) + 6(w+z) = (w + 6)(w+z) 2. a2b2 + a2b + b+1

2. 2ab + bc + 6ad + 3cd

= (a2b2 + a2b) + (b+1)

=

= a2b (b + 1) + 1(b+1) = (a2b +1)(b+1) 3. 2x2 - 4xy + 6y – 3x

3. de – de2 + 7de – 7d

=(2x2 - 4xy) + (6y – 3x)

=

= 2x (x -2y) + 3 (2y- x) = 2x(x – 2y) – 3(x - 2y) = (2x - 3)( x - 2y) 4. ab + bc – ad – dc

4. 10 + 3ab – 15a – 2b

= (ab + bc) – (ad + dc)

=

= b(a+c) – d(a+c) = (b– d)(a+c) Algebraic Expressions

89

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Common Errors Errors 2m − 2 1. 6m 2 =

2.

Correct Steps 2m − 2 1. 6m 2

m−2 3m 2

x2 − 9 x−3 =

=

2(m − 1) 2 × 3× m2

=

m −1 3m 2

x2 − 9 x−3

2.

( x − 3)( x − 3) ( x − 3)

=

( x − 3)( x + 3) ( x − 3)

=x+3

= x–3 3.

3. x2 – 9

x2 – 9 = x2 - 32

= (x + 9 ) ( x – 9)

= (x + 3 ) ( x – 3) 4. y2 - 62

4. y2 - 62

= ( y – 6 )( y + 6)

= ( y – 6 )2

J) Factorizing & Simplifying Algebraic Expressions * Algebraic Fractions are fractions with either its numerator or denominator or both having algebraic expressions Examples : Algebraic Expressions

x p − q 2a , , y + 1 x 2 4 − 3b 90

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I) Simplifying algebraic Expressions * divide the numerator and denominator by their common factors. * factorizing the numerator or denominator or both and then divide the numerator and denominator by their common factors. Example 4a 2 b 1. 2ab 2 4× a× a×b = 2× a×b×b 2 × a 2a = = b b

Exercise 5rs 1. 10r 2 s =

− 12m 3 n 2. 20m 2 n 2 − 4 × 3× m × m × m × n = 5× 4× m× m× n × n − 3× m 3m =− = 5× n 5n

2.

3.

b+c b2 − c2 (b + c) 1 = = (b − c)(b + c ) b − c

x2 − 9 4. 2x 2 − 6x x 2 − 32 = 2× x × x − 2× 3× x ( x − 3)( x + 3) = 2 x ( x − 3) Algebraic Expressions

3.

21c 2 de 3 14cde 2 =

3−d 9−d2 =

e2 − f 2 4. 3e − 3 f =

91

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=

x+3 2x

ii) Addition & Subtraction of Algebraic Expressions a) Algebraic Fractions with same denominator Example Exercise 2x 4 rt 5s − − 1. 1. y y 9k 9k = 2x − 4 = y 2.

2m − 1 m + 4 − 3n 3n (2m − 1) − (m + 4) = 3n 2m − 1 − m − 4 = 3n 2m − m − 1 − 4 = 3n m−5 = 3n

2.

x − 4 3x + 2 − 6 xy 6 xy =

b) Algebraic Fractions with different denominator Example Exercise 1 2 4x y + 1. + (LCM = 5b) 1 b 5b 2 5 = 1× b 2 = + 5b 5b b+2 = 5b 2.

5y 3 − ( LCM =4x2y) 2 2 xy 4x 5 y × y 3 × 2x − = 4x 2 y 4x 2 y

2.

5 y 2 − 6x = 4x 2 y 3.

2d − 3 5d + 1 − 2a 3b

Algebraic Expressions

(LCM = 6ab) 92

4q 5q − 2 a b 2ab 2 =

Module PMR

3b × (2d − 3) 2a × (5d + 1) − 6ab 6ab 6bd − 9b 10ad + 2a − = 6ab 6ab 6bd − 9b − (10ad + 2a ) = 6ab 6bd − 9b − 10ad − 2a = 6ab =

3.

ab + 1 2 − a − 4r 18rs =

iii)Multiplication and Division of Algebraic Expressions a) Multiplication 2 algebraic fractions involving 2 types : * Denominator with one term Example 1. 2 p 3q × 7 5a 2 p × 3q = 7 × 5a 6 pq = 35a 2.

6m 2 4n 2 × 2n 3m 2 × 3× m × m 2 × 2 × n × n × = 2× n 3× m 2× m× 2× n = 1 = 4mn

Algebraic Expressions

Exercise 7 2 × 1. 3r 9 y =

2.

3.

3x 18 y × 2 6 yz 15 x 2 =

p+r 3 × 12 x q

93

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* denominator with two terms 1.

Example x2 − y2 x × 2x x− y ( x − y )( x + y ) x . = × 2x ( x − y) x+ y = 2

2.

2x 2 + x x 2 − y 2 × x+ y 2 xy + y x(2 x + 1) ( x − y )( x + y ) × = x+ y y (2 x + 1) x( x − y ) = y

Exercise 3x x−3 × 1. 2 x − 9 6x 2 =

2.

2x − 2 x + 3 × x 2 − 9 4x − 4 =

.

Division of Algebraic Fractions * Denominator with one term Example a 2c a e ae ÷ = × = 1. d e d 2c 2cd

2.

3.

m − n 2( m − n ) ÷ 4c 2c ( m − n) 2c × = 2(m − n) 2 4c 1 = 4 x 2 − 16 y 2 x + 4 y ÷ xy 2x 2 2 2 x − (4 y ) x + 4y ÷ = 2 xy 2x ( x − 4 y )( x + 4 y ) x× y × = 2× x× x ( x + 4 y)

Algebraic Expressions

Exercise 5 10 z 1. ÷ 2 x x =

2.

3.

94

a + b 2a + 2b ÷ cd 3c 2 d =

3b − 3a a 2 − b 2 ÷ 2d 4d 2 =

Module PMR

=

y( x − 4 y) 2x

.

m − 5 m 2 − 25 ÷ 4. 4 pq 6q

*Denominator with 2 terms Example 1. 2p p ÷ 2 q+2 q −4 2 p q 2 − 22 × q+2 p 2p (q − 2)(q + 2) = × (q + 2) p = 2(q − 2) =

2.

ab 2 2a ÷ 2 ( a − b) ( a − b 2 ) ab 2 2a ÷ (a − b) (a − b)(a + b) ab 2 (a − b)(a + b) × = ( a − b) 2a 2 b ( a + b) = 2

Exercise 5 10m ÷ 1. n − 3 3n − 9 =

3g 2 27 gh ÷ 2. 2 2 y−x y −x =

=

Algebraic Expressions

3.

95

s 2 + st rs + rt ÷ 2 w − 3 12 − 8w =

Module PMR

3,

7 x 2 − 14 x 2 x − 4 ÷ 2x + 2 x +1 1 7 x ( x − 2) ( x + 1)1 × = 2( x + 1)1 2( x − 2)1 7x ×1 = 2× 2 7x = 4

d 2 − e 2 ( d − e) 2 4. ÷ 12b − 6a 3a − 6b =

5.

ab 2 2a ÷ 2 ( a − b) ( a − b 2 ) =

Common Errors Errors 3b + 6 1. 9b 3b1 + 6 = 9b3 1+ 6 7 = = 3 3 2.

3.

Correct Steps 3b + 6 1. 9b 3(b + 2) = 9b3b b+2 = 3b

1 2 + x 3x 1+ 2 = x + 3x 3 = 4x

2.

2 p − 3q 9q − 6 p (2 p − 3q )1 1 = = 3(3q − 2 p)1 3

Algebraic Expressions

3.

96

1 2 + x 3x 1× 3 2 + = 3x 3 x 3+ 2 5 = = 3x 3x 2 p − 3q 9q − 6 p ( 2 p − 3q )1 = − 3(2 p − 3q )1

=-

1 3

Module PMR

4.

5.

1 h −1 − hk h 1 (h − 1)(×k ) = − hk h(×k ) 1 − kh − k = hk

4.

2x ÷ xy y 2x = × xy y

1 h −1 − hk h 1 (h − 1)(×k ) = − hk h(×k ) 1 − kh + k = hk 2x ÷ xy y 2x 1 2 = × = 2 y xy y

5

= 2x 2 6.

m + 4 2x − m m 4 2x = − m m 4 − 2x 2x = = m m

6

m + 4 2x − m m m + 4 − 2x = m

Questions based on PMR Format (A) Simplify each of the following expressions : 1) 4a – (a – 5) 11. (a – 3)2

2) 10q + ( -6q) -5

12 ( 3x + 2)2

3) 6p – ( -3p) – 2p

13. (5d – t)2

4) 4a – a( b+4)

14. (x – 2)2 – x( x -6)

Algebraic Expressions

97

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5) -5m – 4(m – 2)

15. ( 2y + 3)2 – ( 5y - 2)

6) 6b – (b +3)

16. ( 3w –z)2 + z(2w –z)

7) 5x – 3(2 - x)

17) (k-2)2- 8 + 3k

8) 4k(k – 3m) – 3m(m – 4k)

18) ( 6s -1)2 – ( 4s + 1)

9) 3(x –y ) – 2 ( y – x)

19) 2 ( 3y- 4) + ( y -5)2

10) -3 ( c – d) + 2 ( 4c -2d)

20) (2p +q)2 - q(4p – 2q)

(B) Factorise completely each of the following expressions : 1. 12xy – 4x2 11. 4x -3y –xy – 12

Algebraic Expressions

98

Module PMR

2. 6e – 18ef

12, a2 b2 + a2b + b + 1

3. 4x2 -100

13. 2m2 –m + 2mn

4, 75 – 3m2

14 9c2 – 100d2

5. 3y + 12

15 uv + wv –ux –wx

6. 20 – 5x2

16 ab+ bc –ad-cd

7. 3st – 15st2u

17. k2-14k + 49

8. 36x2 – 81y2

18. g2 -12g + 36

9. m3 – 9m

19. 3x-4y-6wx+8wy

10. 4p2 -1

20. 2pq- 6pz – 3rq + 9rz

Algebraic Expressions

99

Module PMR

(C) Express each of the following expressions as a single fraction in its simplest form 2 x p 1− p − − 1. 9. 3 x 12 3m m

2.

4q 3 − p pq

10

2m − 3 m + 2 − 4 6

3.

4 1− s − s 2s 2

11

2  m−3 −  3m  12m 2 

4.

5 x −1 − 2w t

12.

8 1 − 4w − 5n 10n

5.

x+3 2− y − 2e 4

13.

3( p + 1) 6 − n − 3np n

6.

p 1− p + 3f f

14.

7 m−4 − 12m 4m 2

Algebraic Expressions

100

Module PMR

7

1 2z − 1 − 4n 8n 2

15.

8.

5p p2 + 6 − 6q 12 pq

16.

9 y − y ( y + 3) y + 3

2 4−n − 3n 9n 2

(D) Expand each of the following expressions 1. 2 (m+1) 10. (p + 2z )( p – x)

2. 3b (b – 3)

11 (n -7)2

3 -2a ( x – 4)

12. ( r – t)2 -4rt

4. 2k2 ( k – 7)

13 (4m -2)2 + 7m

5. – 5x ( x – 2y)

14 (a+ 2d)( a+ 2d)

Algebraic Expressions

101

Module PMR

6. 2e( 4e – f + 7)

7

15 (3ª +b)(2a- 2c)

16

2 s (6 x − 12 y + 9) 3

( x – 3y)( x + 3y)

8. -6pq(2pq + 4p – 3q)

17 . (2a + 1)( b- 3)

9 4 ( - 3s + 5h)

18. (x -2)( y + 3)

PMR past year questions 2004 1. Simplify (3x-1)2 –(7x + 4)

(2 marks)

2. Factorise completely a) 9xy -3x2

b) p2 – 6(p+1) – (8 –p)

( 3 marks)

Algebraic Expressions

102

Module PMR

1   1− p  3  2  − 3. Express as a single fraction in its simplest form 2m  mp      ( 3 marks)

2005 4. Simplify (2p- q)2 + q(4p –q)

5.

( 2marks)

Factorise completely each of the following expressions : (a) 4e – 12ef b) 3x 2 - 48 ( 3 marks)

5. Express

1 m+2 − as a single fraction in its simplest form. 2m 6m 2 ( 3 marks)

2006 6. Factorise completely 50 – 2m2

Algebraic Expressions

103

(2 marks)

Module PMR

7. Simplify 3 (2p -5) + (p – 3)2

8. Express

( 2 marks)

1 5 − 2v − as a single fraction in its simplest form 5m 15mv (3 marks)

2007 9. Factorise completely each of the following expressions : b) 12 – 3x2

a) 2y + 6

( 3 marks)

10. Expand each of the following expressions : (a) q(2 + p) (b) ( 3m –n)2 ( 3 marks)

11. Express

Algebraic Expressions

5 2 − 3w + as a single fraction in its simplest form 3n 6n (3 marks)

104

Module PMR

2008 12. Simplify 2p – 3q – (p + 5q)

( 2 marks)

13. Expand each of the following expressions : (a) 2g ( 5 –k)

(b) ( h – 5)(3h + 2) ( 3 marks)

14. Express

1 n−4 − as a single fraction in its simplest form n 3n ( 3marks)

CHAPTER 7 : ALGEBRAIC EXPRESSIONS ANSWERS A unknown 1. Number of books 2. number of monkeys 3. unknown : y Object : z 4. unknown : k Object : a

3) 0.7 4) 1 (ii) 1) unlike term 2) like term 3) unlike term 4) unlike term C) Algebraic Expressions i) Number of term 1) 3 2) 4 3) 5 ii) simplify Algebraic Exp. 1) 7x – 8y 2) 3z – 11w

B Algebraic terms (i) 1) -3 2 2) 5 Algebraic Expressions

105

Module PMR

3) 9r + 7s 4) -2k-5 5) -5t+7s 6) 2s -1 7) -18w

H. Expanding double brackets 1. ab + a -2b -1 2. 3m2-mn + 9m -3n 3. -6st – 8s – 15t -20 4. a2 -6a + 9 5. 9m2 -6mn +n2 6. 25x2+20x + 4 7. y2+8dy+16d2

D) Alg Terms in two or more terms * Identify coefficient of unknown -3b , -3c, -3, -3b2 E) Multiplication & division of alg terms i) find product of 2 alg terms 1) a3b2 2) -6xy2z 3) 3ab3c4 4) 56p4q2r3 5) 6w3z4 2 2 2 2 6) - p q r 5 7) 21m 4 h 2 k 5

I. Factorization 1. 6( a – 4c) 2. 2m2( 2m -3) 3. 2x ( 4a + 2b –c) 4. xyz( x -y) 5. 3st (t-5w) 6. 2yz( 1- 2z +3x)

i) expressions which consist of 2 terms 1) (w – 5)(w + 5) 2) 5(x – 1)(x + 1) 3) 3(2d-5)(2d +5) 4) (6c -10e)(6c +10e)

ii) find quotient of 2 alg. Terms 2ab 3 pqz 1) 3) − r 3d 4x 2) 4) -3rt y iii) Multiplication &Division of alg terms 1. 16p2r2 2 18k 3. -4ab4 F. Computation involve Alg Exp. 1. 40m – 16 2. -2a – 6b 3. -8p + 21 4 10 + 3t 5 2s - 33 6 -3pq+2qr –pqr 7 x – 3 – 5y 8. -3p2 9. 6u2-12 G. Expanding single Brackets 1. wy + y2 2. -15ef – 10eg 3 4r – 3rs + rt 4 4xyz – 2xyw + x2y2 Algebraic Expressions

ii) expressions which consist of 3 terms 1) (a + b)2 2) (2x – 5)2 3) ( 3e – 2f)2 iii) expressions which consist of 4 terms 1) (q +s)(p + r) 2) (b+ 3d)(2a+c) 3) (de – 7e)(1- e) 4) (5 –b)(2 – 3a ) J) Factorising & simplifying Alg Expressions. i) Simplifying alg Expressions 1 1. 2r 3ce 2 2 1 3. 3+ d 106

Module PMR

e+ f 3 ii) Addition & Subtraction of alg Exp. a) Alg Fraction with same denominator rt − 5s 1. 9k 3− x 2 3 xy b)Alg Fraction with different denominator 10 x + y 1. 5 8bq − 5aq 2. 2a 2 b 2 9abs + 9b − 4 + 2a 3. 36rs 4

iii) Mul & division of Alg Exp a) Multiplication of 2 alg fractions * Denominator with one term 14 1. 27 ry 3 2. 5 xz 2 p+r 3. 4 xq * denominator with 2 terms 1 1. 2 x( x + 3) 1 2 2( x + 3) b) Division of Alg Fractions * Denominator with one term x 1. 2z 3c 2. 2 3 3. − 2d ( a + b) Algebraic Expressions

4.

3 2 p (m + 5)

* Denominator with 2 terms 3 1. 2m g 2. 9h( y + x ) 4s 3. − r d +e 4 − d −e b 2 ( a + b) 5. 2

Questions based on PMR Format A. Simplify expressions 1. 3a+ 5 2. 4q -5 3. 7p 4. –ab 5. -9m+8 6. 5b -3 7. 8x – 6 8. 4k2 - 3m2 9. 5x – 5y 10. 5c – d 11 a2 - 6a + 9 12 9x2+ 12x +4 13 25d2 -10dt + t2 14 2x + 4 15 4y2 + 7y + 11 16 9w2 - 4wz 17 k3 –k -4 18 36s2 – 16s 19 y2 -4y + 17 20 4p2 + 3q2 B. Factorise Expressions 1 4x( 3y – x) 2. 6e( 1 – 3f) 107

Module PMR

3 4. 5. 6. 7. 8 9 10 11 12 13 14 15 16 17 18 19 20

(2x – 10)( 2x + 10) 3 (5 –m)(5+m) 3( y + 4) 5(2-x)(2+x) 3st(1 – 5tu) 9(2x -3y)(2x+3y) m(m-3)(m+3) (2p-1)(2p+1) (x+3)(4-y) (a2b+1)(b+1) m(2m-1+2n) (3c-10d)(3c+10d) (v-x)(u+w) (b-d)(a+c) (k -7)2 (g – 6)2 (3x -4y)( 1-2w) (2p-3r)(q-2)

C) Express expressions as a single fraction in its simplest form 8 − x2 1. . 12 x 4q 2 − 3 2. pq 9s − 1 3. 2s 2 5t − 2 wx + 2 w 4. 2 wt 2 x + 6 − 2e + ey 5. 4e p + 3 − 3p 6. 3f 2n − 2 x + 1 7. 8n 2 9 p2 − 6 8. 12 pq PMR Past Year Questions

9. 10. 11. 12. 13 14. 15. 16.

D Expand expressions 1. 2m+2 2. 3b2 -9b 3. -2ax+ 8x 4.. 2k3 – 14k2 5. -5x2+ 10xy 6. 8e2 -2ef +14e 7. 4sx -8sy +6s 8. -12p2q2 -24p2q +18pq2 9. -12s +20h 10. p2 –px + 2pz – 2zx 11. n2 -14x +49 12. r2 -6rt +t2 13. 16m2 -9m + 4 14, a2 +4ad + 4d2 15. 6a2 -6ac+2ab-2bc 16. x2 + 9y2 17. 18.

2004 1. 9x2 -13x -3 2. a) 3x(3y-x) b) p2- 7p -14 2 p −1 3. mp Algebraic Expressions

4p −3 m 4m − 13 12 7m + 3 12m 2 15 + 4 w 10n 3 − 15 p + 3 pn 3np m+3 3m 2 3− y y 7n − 4 9n 2

108

2ab- 6ª + b -3 xy +3x -2y -6

Module PMR

2005 4. 4p2 5. a) 4e(1-3f) 6. m −1 7. 3m 2

b) 3(x-4)(x+4)

2006 6. 2(5 –m)(5 +m) 7. p2 – 6 v −1 8. 3mv 2007 9. a) 2(y+3) 10. a) 2q+pq 4−w 11. 2n

b) 3(2-x)(2+x) b) 9m2- 6mn +n2

2008 12. p -8q 13. a) 10g -2gh b) 3h2 -12h -10 7−n 14. 3n

Algebraic Expressions

109