Module PMR
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 Exercise1. The teacher gives some
pencils to the studentsSolution : The unknown is the number of pencils
2. There are x students in m class Solution : x is unknown m is object
1. I bought some books Solution : ……………………………
2. There are many monkeys in the garden. Solution : …………………………….
3. Azman bought y durian in z shop 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 = 4 x y = y + y + y + y 4y
Number unknown * Identify coefficients in given algebraic term - Coefficient is the number that multiply the unknown
Example Exercise1) 7m : coefficient of m is 7
2) coefficient of r is
3) – y : coefficient of y is -1
1) -3z : coefficient of z is……………
2) : coefficient of x is …………..
3) 0.7 h : coefficient of h is …………
4) p : coefficient of p is …………..
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(ii) Like and Unlike Algebraic Terms * Like term : terms with the same unknowns * Unlike terms : terms with different unknowns.
Example Exercise
1. 3m and -4m Like term 2. 4x and ¼ x (same unknown)3. 0.9z and 5z
1. 2w and 8h unlike term2, -5f and ½g (different unknown)3. 1.2q and 3.5g
Determine whether each of the following pairs of algebraic terms are like term / unlike term
1. 6s , - t : ……………………
2. , 8y : ……………………..
3. 19 d , 19e : ……………………
4. , 4e : ………………………..
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 ExerciseDetermine the number of terms in the algebraic expressions below :
1. 3x + 6y : 2 terms
2. 7p + 5q – 9 : 3 terms
3. w – 2z – 8y + 1 : 4 terms
Determine the number of termsin the algebraic expressions below :
1. 6m + 8n – 9 : …………………..
2. 3b + 2e – 10b -5e : …………….
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 78
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Examples Exercise1. 6m – 2n + 4m – 5n = 6m + 4m -2n -5n (group like terms)
= 10m – 7n
2. -7x + 4y + 3y + 2x = -7x + 2x + 4y + 3y = -5x + 7y
3. ( 12a – 4b) + ( 5a + 7b) = 12a+ 5a – 4b + 7b = 17a + 3b
4. ( 9q + 2p) – ( 4q – 6p) = 9q + 2p – 4q + 6p = 9q – 4q + 2p + 6p = 5q + 8p
5. 8x – ( - 4x) + x = 8x + 4x + x = 13x
6. -3c –(- d) +(-2d) = -3c +d – 2d =-3d -d
.1. 2x – 7y + 5x – y =
2. 11z -3w - 8z- 8w =
3. ( 6r + 9s) + ( 3r – 2s) =
4. ( 5k -3) – ( 7k + 2) =
5. (2t +4s) – (7t – 3s) =
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 ExerciseIn 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
1. in the term -3ab2c* the coefficient of abc =………………..* the coefficient of ab2 = ………………..* the coefficient of ab2c =……………..* the coefficient of ac = ……………….
Algebraic Expressions 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 Exercise1. 2ab x 4b2c = 2 x a x b x 4 x b x b x c = 2 x 4 x a x b x b x b x c = 8 x a x b2 x c = 8ab3c
2. 4m2 x ½ mn2
= 4 x ½ x m x m x m x n x n = 2 x m3 x n2
= 2 m3 n2
Exercise
6. =
7. (-3m2hk3)x (-7m2hk2) =
1. ab x a2b
=
2. 3xy x (-2 yz)
=
3.. 6ab2c x (½ bc3)
=
4. (-8p3qr) x ( -7pqr2)
=
5.
=
Algebraic Expressions 80
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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 Exercise
1.
=
= 2y
2. 12m2n ÷ 3mn
=
= 4m
3. -5cd2e ÷ 15c2de
=
=
1. 24pq2z ÷ 8qr =
2.
=
3. 12abc ÷ (-18cd) =
4. (- 18sr3t2) ÷ 6sr2t =
iii) Multiplication and Division involving algebraic terms
Example Exercise1. 4p x 6q2 ÷ 3pq
= 8q
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
1. 6p2qr ÷ 3pq x 8pr =
2.
=
3. 10a2b3 x (-2b2c) ÷ 5abc
Algebraic Expressions 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 Exercise1. 3 ( 2a –b) = 3 x 2a – 3 x b = 6a – 3b
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
7. 5(2x – 1) -
= 10x -5 – (3x - 1) = 10x -5 -3x + 1 = 10x -3x -5 + 1 = 7x -4
1. 8 ( 5m -2) =
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.
=
8.
=
9.
=
Algebraic Expressions 82
2
6
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Common Errors
Errors Correct Steps1. 7pq x 3pq = 21 pq
2. 2 ( 4e – 3 d) = 8e – 3d
3. (6de2 – 4ef) ÷ 2e = 3de – 4ef
4. (x – 4y) – ( 2x + y) = x – 4y – 2x + y = x -2x -4y + y = -x-3y
5. -2p ( pq – 3) = - 2pq – 6p
6. 10abc – 4 abc = 6
7. 3a +6b – 8a – 3b = 3a + 8a - 6b -3b = 11a – 9b
8. ( - 4rs2t) x 5r3st2
= (-4) x 5 x r x r3 x s2 x s x t x t2 = 20r3s2t2
9. -5s - ( 3t – 2)
= +15st -10s
1. 7pq x 3pq = 7 x 3 x p x p x q x q = 21 p2q2
2. 2( 4e – 3d) = 2 x 4e – 2 x 3d = 8e – 6d
3. (6de2 – 4ef) ÷ 2e = 6de 2 ÷ 2e – 4ef ÷ 2e = 3de – 2f
4. (x – 4y) – ( 2x + y) = x – 4y – 2x - y = x -2y -4y - y = - x - 5y
5. -2p ( pq – 3) = - 2p2q + 6p
6. 10abc – 4 abc = 6abc
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 = -20r4s3t3
9. -5s - ( 3t – 2)
= -5s -3t +2
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
Example Exercise1. 2p (p – 3q) = 2p x p – 2p x 3q = 2p2 – 6pq
2. -4b(2a + b) = -4b x 2a -4b x b = -8ab – 4b2
3. (6a – 9c)
= x 6 2a - x 9c
= 4ab -6bc
1. y ( w + y) =
2. -5e ( 3f + 2g) =
3.
=
4. xy ( 4z – 2w + xy) =
5.
=
6. - 7ab(2a – 4b + c) =
Algebraic Expressions 84
* p(q + r)= p q + p r = pq + pr
3
Module PMR
(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 Exercise1. (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
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 Correct Steps1. 2x(x-3)
= 2x2 -3
2. (a+b)2
= a2 + b2
3. ( a – b)2
= a2 – b2
4 2m2(3m2n – 4 mn3)
= 6m4n – 4m2n3
5. -4 ( 3de – 2rst2)
= - 12de – 2rst2
6. ( x -3)2
= x2 – 9
7. 4a2 –(a + b)2
= 4a 2 –a2 + b2
= 3a2 + b2
8. (2x -3)(x + 4)
= 2x( x+4) – 3 (x + 4)
= 2x2 + 8x – 3x + 12
1. 2x(x-3)
= 2x2 – 6
2. (a+b)2
= a2 +2ab + b2
3.. ( a – b)2
= a2 – 2ab + b2
4. 2m2(3m2n – 4 mn3)
= 6m4n – 8m3n3
5. -4 ( 3de – 2rst2)
= - 12de + 8rst2
6. ( x -3)2
= x2 – 6x + 9
7. 4a2 –(a + b)2
= 4a 2 –( a2 +2ab + b2 )
= 4a2 – a2 - 2ab - b2
= 3a2 -2ab –b2
8. (2x -3)(x + 4)
= 2x( x+4) – 3 (x + 4)
= 2x2 + 8x – 3x - 12
Algebraic Expressions 86
Module PMR
= 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 Exercise1. st – sr = s( t – r )
2. 4m + 12mn – 16m2
= 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)
4. 10mn – 15m2
= 5 x 2 x m x n – 5 x 3 x m x m = 5m ( 2n – 3m )
1. 6a – 24c =
2. 4m3 – 6m2
=
3. 8ax + 4bx – 2cx =
4. x2yz – xy2z =
5. 3st2 – 15 stw =
6. 2yz – 4yz2 + 6xyz =
Algebraic Expressions 87
ab – ac = a ( b – c) a = common factor
Module PMR
*Factorize an expression by using the difference between 2 squares i) expressions which consist of 2 terms :
Example Exercise1 9 – a2
= 32 – a2
= ( 3-a)(3+a)
2. 4x2 – 25y2
= 22 x2 – 52 y2 = ( 2x)2 – (5y)2
= ( 2x – 5y)( 2x + 5y)
3. 8g2 – 18h2
= 2 ( 4g2 – 9h2 ) = 2 [ ( 2g)2 – ( 3h)2 ] = 2 (2g -3h) (2g + 3h)
1. w2 – 25 =
2. 5x2 -5 =
3. 12d2 – 75 =
4. 36c2 -100e2
=
ii) expressions which consist of 3 terms
Example Exercise1. 9x2 + 6xy + y2
= (3x2) + 2 (3x)(y) + y2
= ( 3x + y)2
2. p2 – 4pq + q2 = p2 – 2(p)(q) + q2
= (p – q)2
3. 16p2 – 24pq + 9q2
1. a2 + 4ab + b2
=
2. 4x2- 20x + 25 =
Algebraic Expressions 88
a2 – b2 = ( a – b)( a + b)
a2 + 2ab + b2 = (a + b)2
a2 - 2ab + b2 = (a - b)2
Module PMR
= (4p)2 – 2(4p)(3q) +(3q)2
= (4p –3q)2 3. 9e2 – 12 ef + 4f2
=
iii) expressions which consist of 4 terms
Example Exercise1. w2 + wz + 6w + 6z
= (w2 + wz) +( 6w + 6z)
= w(w+z) + 6(w+z)
= (w + 6)(w+z)
2. a2b2 + a2b + b+1
= (a2b2 + a2b) + (b+1)
= a2b (b + 1) + 1(b+1)
= (a2b +1)(b+1)
3. 2x2 - 4xy + 6y – 3x
=(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
= (ab + bc) – (ad + dc)
= b(a+c) – d(a+c)
= (b– d)(a+c)
1. pq + qr + ps + rs
=
2. 2ab + bc + 6ad + 3cd
=
3. de – de2 + 7de – 7d
=
4. 10 + 3ab – 15a – 2b
=
Algebraic Expressions 89
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)
Module PMR
Common Errors
Errors Correct Steps
1.
=
2.
=
= x – 3
3. x2 – 9
= (x + 9 ) ( x – 9)
4. y2 - 62
= ( y – 6 )2
1.
=
=
2.
=
= x + 3
3. x2 – 9
= x2 - 32
= (x + 3 ) ( x – 3)
4. y2 - 62
= ( y – 6 )( y + 6)
J) Factorizing & Simplifying Algebraic Expressions
* Algebraic Fractions are fractions with either its numerator or
denominator or both having algebraic expressions
Examples :
Algebraic Expressions 90
Module PMR
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 Exercise
1.
=
=
2.
=
=
3.
=
4.
=
=
=
1.
=
2.
=
3.
=
4.
=
Algebraic Expressions 91
Module PMR
ii) Addition & Subtraction of Algebraic Expressionsa) Algebraic Fractions with same denominator
Example Exercise
1.
=
2.
=
=
=
=
1.
=
2.
=
b) Algebraic Fractions with different denominatorExample Exercise
1. (LCM = 5b)
= +
=
2. ( LCM =4x2y)
=
=
3. (LCM = 6ab)
=
=
=
1
=
2.
=
Algebraic Expressions 92
Module PMR
= 3.
=
iii)Multiplication and Division of Algebraic Expressions
a) Multiplication 2 algebraic fractions involving 2 types : * Denominator with one term
Example Exercise
2.
=
=
= 4mn
1.
=
2.
=
3.
* denominator with two terms
Example Exercise
.
2.
1.
=
2.
=
Algebraic Expressions 93
1.
1.
Module PMR
=
=
.
Division of Algebraic Fractions
* Denominator with one term
Example Exercise
1.
2.
=
=
3.
=
=
=
.
1.
=
2.
=
3.
=
4.
Algebraic Expressions 94
Module PMR
*Denominator with 2 terms
Example Exercise
2.
=
=
=
3,
=
=
=
1.
=
2.
=
3.
=
4.
=
5.
=
Algebraic Expressions 95
1.
Module PMR
Common Errors
Errors Correct Steps
1.
=
=
2.
=
=
3 .
= =
4.
1.
=
=
2.
=
=
3.
= = -
4.
Algebraic Expressions 96
5 5
6
Module PMR
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)
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
Algebraic Expressions 97
6.
Module PMR
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
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
Algebraic Expressions 98
Module PMR
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
(C) Express each of the following expressions as a single fraction in its simplest form
1. 9.
2. 10
Algebraic Expressions 99
Module PMR
3. 11
4. 12.
5. 13.
6. 14.
7 15.
8. 16.
Algebraic Expressions 100
Module PMR
(D) Expand each of the following expressions1. 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)
6. 2e( 4e – f + 7) 15 (3ª +b)(2a- 2c)
7 16 ( x – 3y)( x + 3y)
8. -6pq(2pq + 4p – 3q) 17 . (2a + 1)( b- 3)
Algebraic Expressions 101
Module PMR
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)
3. Express as a single fraction in its simplest form
( 3 marks)
2005
4. Simplify (2p- q)2 + q(4p –q) ( 2marks)
Algebraic Expressions 102
Module PMR
5. Factorise completely each of the following expressions : (a) 4e – 12ef b) 3x - 48 ( 3 marks)
5. Express as a single fraction in its simplest form.
( 3 marks)
2006
6. Factorise completely 50 – 2m2 (2 marks)
7. Simplify 3 (2p -5) + (p – 3)2 ( 2 marks)
8. Express as a single fraction in its simplest form
(3 marks)
2007Algebraic Expressions 103
Module PMR
9. Factorise completely each of the following expressions :
a) 2y + 6 b) 12 – 3x2 ( 3 marks)
10. Expand each of the following expressions : (a) q(2 + p) (b) ( 3m –n)2 ( 3 marks)
11. Express as a single fraction in its simplest form
(3 marks)
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)
Algebraic Expressions 104
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14. Express as a single fraction in its simplest form
( 3marks)
CHAPTER 7 : ALGEBRAIC EXPRESSIONSANSWERS
A unknown 1. Number of books 2. number of monkeys 3. unknown : y Object : z 4. unknown : k Object : a
B Algebraic terms (i) 1) -3
2)
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 3) 9r + 7s 4) -2k-5 5) -5t+7s 6) 2s -1 7) -18w 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
6) -
7) 21m ii) find quotient of 2 alg. Terms
1) 3)
2) 4) -3rt
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
Algebraic Expressions 105
Module PMR
2. -15ef – 10eg 3 4r – 3rs + rt 4 4xyz – 2xyw + x2y2
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
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) 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.
2
3.
4
ii) Addition & Subtraction of alg Exp.
a) Alg Fraction with same denominator
1.
2
b)Alg Fraction with different denominator
1.
2.
3.
iii) Mul & division of Alg Exp a) Multiplication of 2 alg
fractions * Denominator with one term
1.
2.
3.
* denominator with 2 terms
1.
2
b) Division of Alg Fractions * Denominator with one term
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Algebraic Expressions 106
Module PMR
* Denominator with 2 terms
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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) 3 (2x – 10)( 2x + 10) 4. 3 (5 –m)(5+m) 5. 3( y + 4)
6. 5(2-x)(2+x) 7. 3st(1 – 5tu) 8 9(2x -3y)(2x+3y) 9 m(m-3)(m+3) 10 (2p-1)(2p+1) 11 (x+3)(4-y) 12 (a2b+1)(b+1) 13 m(2m-1+2n) 14 (3c-10d)(3c+10d) 15 (v-x)(u+w) 16 (b-d)(a+c) 17 (k -7)2
18 (g – 6)2
19 (3x -4y)( 1-2w) 20 (2p-3r)(q-2)
C) Express expressions as a single fraction in its simplest form
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Algebraic Expressions 107
Module PMR
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. 2ab- 6ª + b -3 18. xy +3x -2y -6
PMR Past Year Questions
2004
1. 9x2 -13x -32. a) 3x(3y-x)
b) p2- 7p -14
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2005
4. 4p2
5. a) 4e(1-3f) b) 3(x-4)(x+4)6.
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2006
6. 2(5 –m)(5 +m)7. p2 – 6
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2007
9. a) 2(y+3) b) 3(2-x)(2+x)10. a) 2q+pq b) 9m2- 6mn +n2
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2008
12. p -8q13. a) 10g -2gh b) 3h2 -12h -10
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Algebraic Expressions 108
Module PMR
Algebraic Expressions 109