chapter 9: algebraic expressions and identities

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Chapter 9: Algebraic Expressions and Identities

Exercise 9.1 (Page 140 of Grade 8 NCERT)

Q1. Identify the terms, their coefficients for each of the following expressions.

(i) 25 3xyz zy−

(ii) 21 x x+ +

(iii) 2 2 2 2 2 24 4x y x y z z− +

(iv) 3 pq qr rp− + −

(v) 2 2

x yxy+ −

(vi) 0.3 0.6 0.5a ab b− +

Difficulty level: Easy

Known:

Expressions

Unknown:

Terms and their coefficients

Reasoning:

The numerical factor of a term is called its numerical coefficient or simply coefficient.

Solution:

The terms and the respective coefficients of the given expressions are as follows.

- Terms Coefficients

(i) 25

- 3

xyz

zy

5

3−

(ii)

2

1

x

x

1

1

1

(iii)

2 2

2 2 2

2

4

4

x y

x y z

z

−

4

4

1

−



(iv) 3

pq

qr

rp

−

−

3

1

1

1

−

−

(v)

2

2

x

y

xy−

1

2

1

2

1−

(vi) 0.3

0.6

0.5

a

ab

b

−

0.3

0.6

0.5

−

Q2. Classify the following polynomials as monomials, binomials, trinomials.

Which

polynomials do not fit in any of these three categories?

2 3 4 2 2 3

2 2 2

, 1000, , 7 5 , 2 3 ,2 3 4 , 5 4 3 ,

4 15 , , , , 2 2

x y x x x x y x y y y y y x y xy

z z ab bc cd da pqr p q pq p q

+ + + + + + − − + − +

− + + + + +


Known:

Expression

Unknown:

The degree of the expression.

Reasoning:

1. Expression that contains only one term is called a monomial.

2. Expression that contains two terms is called a binomial.

3. Expression containing three terms is a trinomial and so on.

4. An expression containing, one or more terms with non-zero coefficient (with

variables having non-negative integers as exponents) is called a polynomial.

5. A polynomial may contain any number of terms, one or more than one.

Solution:

The given expressions are classified as

Monomials: 1000, pqr

Binomials: 2 2 2 2, 2 3 , 4 15 , , 2 2x y y y z z p q pq p q+ − − + +



Trinomials: 2 37 5 , 2 3 4 , 5 4 3y x y y y x y xy+ + − + − +

Polynomials that do not fit in any of these categories are

2 3 4 , x x x x ab bc cd da+ + + + + +

Q3. Add the following.

(i) , , ab bc bc ca ca ab− − −

(ii) , , a b ab b c bc c a ac− + − + − +

(iii) 2 2 2 2 2 3 4, 5 7 3p q pq pq p q− + + −

(iv) 2 2 2 2 2 2, , , 2 2 2l m m n n l lm mn nl+ + + + +

Difficulty level: Medium

Known:

Expressions

Unknown:

Addition

Reasoning:

Addition will take place between like terms.

Solution:

The given expressions written in separate rows, with like terms one below the other

and then the addition of these expressions are as follows.

(i)

0

ab bc

bc ca

ab ca

−

+ −

+− +

Thus, the sum of the given expressions is 0.

(ii)

a b ab

b c bc

a c ac

ab bc ac

− +

+ − +

+ − + +

+ +

Thus, the sum of the given expressions is ab bc ac+ + .



(iii) 2 2

2 2

2 2

2 3 4

3 7 5

4 9

p q pq

p q pq

p q pq

− +

+ − + +

− + +

Thus, the sum of the given expressions is 2 2 4 9.p q pq− + +

(iv) 2 2

2 2

2 2

2 2 2

2 2 2

2 2 2 2 2 2

l m

m n

l n

lm mn nl

l m n lm mn nl

+

+ + +

+ +

+ + +

+ + + + +

Thus, the sum of the given expressions is 2(2 2 2l m n lm mn nl+ + + + + ).

Q4. (a) Subtract 4 7 3 12a ab b− + + from 12 9 5 3a ab b− + −

(b) Subtract 3 5 7xy yz zx+ − from 5 2 2 10xy yz zx xyz− − +

(c) Subtract 2 24 3 5 8 7 10p q pq pq p q− + − + − from

2 218 3 11 5 2 5p q pq pq p q− − + − +


Known:

Expressions

Unknown:

Subtraction

Reasoning:

Subtraction will take place between like terms.

Solution:

The given expressions in separate rows, with like terms one below the other and

then the subtraction of these expressions is as follows.

(a)

( ) ( ) ( ) ( )

12 9 5 3

4 7 3 12

8 2 2 15

a ab b

a ab b

a ab b

− + −

− + +

− + − −

− + −



(b)

( ) ( ) ( )

5 2 2 10

3 5 7

2 7 5 10

xy yz zx xyz

xy yz zx

xy yz zx xyz

− − +

+ −

− − +

− + +

(c)

( ) ( ) ( ) ( ) ( ) ( )

2 2

2 2

2 2

18 3 11 5 2 5

10 8 7 3 5 4

28 5 18 8 7

p q pq pq p q

p p pq pq p q

p q pq pq p q

− − + − +

− − + − + +

+ + − + − −

+ − + − +





Q1. Find the product of the following pairs of monomials.

( ) ( ) ( ) ( ) ( )3 4,7p 4 , 7 4 , 7 4 , 3 4 , 0i ii p p iii p pq iv p p v p− − −


Known:

Pairs of monomials

Unknown:

Product

Reasoning:

i) By using the distributive law, we can carry out the multiplication term by term.

ii) In multiplication of polynomials with polynomials, we should always look for like

terms, if any, and combine them.

Solution:

The product will be as follows.

( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( )

2

2

3 4

4 7 4 7 28

4 7 4 7 4 7 28

4 7 4 7 4 7 28

4 3 4 3 12

4 0 4 0 0

i p p p

ii p p p p p p p

iii p pq p p q p p q p q

iv p p p p p p p

v p p

= =

− = − = − = −

− = − = − = −

− = − = −

= =

Q2. Find the areas of rectangles with the following pairs of monomials as their

lengths and breadths respectively.

( ) ( ) ( ) ( ) ( )2 2 2, ; 10 , 5 ; 20 , 5 ; 4 , 3 ; 3 , 4p q m n x y x x mn np


Known:

Lengths and breadths of rectangles

Unknown:

Areas of rectangles



Reasoning:

Area of a rectangle = length × breadthSolution:

We know that,

2 2 2 2 2 2

Area of rectangle Length

Area of 1 rectangle

Area of 2 rectangle 10 5 10 5 50


Area of

st

nd

rd

Breadth

p q pq

m n m n mn

x y x y x y

=

= =

= = =

= = =2 2 3

2

4 rectangle 4 3 4 3 12


th

th

x x x x x

mn np m n n p mn p

= = =

= = =

Q3. Complete the table of products.

First monomial

Secondmonomial

→

2x 5y− 23x 4xy− 27x y 2 29x y−

2x 24x5y− 215x y−

23x4xy−

27x y2 29x y−


Known:

Expressions

Unknown:

Product

Solution:

The table can be completed as follows.

First monomial

Secondmonomial

→

2x 5y− 23x 4xy− 27x y 2 29x y−

2x 24x 10xy− 36x 28x y− 314x y 3 218x y−

5y− 10xy− 225y 215x y− 220xy 2 235x y− 2 345x y23x 36x

215x y− 49x312x y− 421x y 4 227x y−

4xy− 28x y− 220xy 312x y− 2 216x y3 228x y− 3 336x y

27x y 314x y 2 235x y− 421x y 3 228x y− 4 249x y 4 363x y−2 29x y− 3 218x y− 2 345x y 4 227x y− 3 336x y 4 363x y− 4 481x y



Q4. Obtain the volume of rectangular boxes with the following length, breadth and

height respectively.

( ) ( ) ( ) ( )2 4 2 25 ,3 ,7 2 , 4 , 8 , 2 , 2 , 2 , 3i a a a ii p q r iii xy x y xy iv a b c


Known:

Length, breadth and height respectively of rectangular boxes

Unknown:

Volume of rectangular boxes

Reasoning: volumeof a rectangular box = length × breadth × height

Solution:

We know that,

( )

( )

( )

2 4 2 4 7

2 2 2

Volume Length Breadth Height

Volume 5 3 7 5 3 7 105

Volume 2 4 8 2 4 8 64

Volume 2 2 2 2

i a a a a a a a

ii p q r p q r pqr

iii xy x y xy xy x y

=

= = =

= = =

= =

( )

2 4 4 4

Volume 2 3 2 3 6

xy x y

iv a b c a b c abc

=

= = =

Q5. Obtain the product of

( ) ( ) ( )

( ) ( )

2 3 2 3 , , , , 2, 4 , 8 , 16

, 2 , 3 , 6 , ,

i xy yz zx ii a a a iii y y y

iv a b c abc v m mn mnp

−

−


Known:

Expressions

Unknown:

Product

Reasoning:

By using the distributive law, we can carry out the multiplication term by term.



Solution:

( )

( ) ( )( )

( )

( ) ( )

2 2 2

2 3 6

2 3 2 3 6

2 2 2

3 2

2 4 8 16 2 4 8 16 1024

2 3 6 2 3 6 36

i xy yz zx x y z

ii a a a a

iii y y y y y y y

iv a b c abc a b c abc a b c

v m mn mnp m n p

=

− = −

= =

= =

− = −





Q1. Carry out the multiplication of the expressions in each of the following pairs.

( ) ( ) ( )

( ) ( )

2 2

2

4 , , , 7

9, 4 , 0

i p q r ii ab a b iii a b a b

iv a a v pq qr rp

+ − +

− + +


Known:

Expressions

Unknown:

Product

Reasoning:




Solution:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( )

2 2

2 2 2 2 2 2 3 2 2 3

2 2 3

4 4 4 4 4

7 7 7 7 7

9 4 4 9 4 4 36

i p q r p q p r pq pr

ii ab a b ab a ab b a b ab

iii a b a b a a b b a b a b a b

iv a a a a a a a

v

+ = + = +

− =

−

+ − = −

+ = + = +

− = −+ =

( ) ( ) ( ) ( ) 0 0 0 0 0pq qr rp pq qr rp+ + = + + =

Q2. Complete the table

--- First expression Second Expression Product

(i) a b c d+ + ---

(ii) 5x y+ − 5xy ---

(iii) p 26 7 5p p− + ---

(iv) 2 24p q 2 2p q− ---

(v) a b c+ + abc ---




Known:

Expressions

Unknown:

Product

Solution:

The table can be completed as follows.

--- First expression Second Expression Product

(i) a b c d+ + ab ac ad+ +

(ii) 5x y+ − 5xy 2 25 5 25x y xy xy+ −

(iii) p 26 7 5p p− + 3 26 7 5p p p− +

(iv) 2 24p q 2 2p q− 4 2 2 44 4p q p q−

(v) a b c+ + abc 2 2 2a bc ab c abc+ +

Q3. Find the product.

(i) ( ) ( ) ( )2 22 26 2 4a a a

(ii) 2 22 9

3 10xy x y

−

(iii) 3 310 6

3 5pq p q

−

(iv) 2 3 4 x x x x


Known:

Expressions

Unknown:

Simplification

Solution:

(i) ( ) ( ) ( )2 22 26 2 22 26 50 2 4 2 4 8a a a a a a a = =

(ii) 2 2 2 2 3 3

5

2 9 2 9 3

3 10 3 10 5xy x y x y x y x y

− − − = =

(iii) 2

3 3 3 3 4 410 6 10 64

3 5 3 5pq p q pq p q p q

− − = = −

(iv) 2 3 4 10 x x x x x =



Q4.

a) Simplify ( )3 4 5 3x x − + and find its values for ( ) ( ) 3,1

2 i x ii x == .

b) ( )2 1 5a a a+ + + and find its values for ( ) ( ) ( )0, 1, 1i a ii a iii a= = = −


Known:

Expression with their corresponding values.

Unknown:

Simplification and its result with their corresponding values

Solution:

( ) 2( ) 3 4 5 3 12 15 3a x x x x− + = − +

( )

( ) ( )

2

2

i For 3,

12 15 3

12 3 15 3 3

108 45 3

66

x

x x

=

= − +

= − +

= − +

=

2

2

3

1(ii) For ,

2

12 15 3

1 112 15 3

2 2

1 1512 3

4 2

153 3

2

156

2

12 15

2

3

2

x

x x

=

= − +

= − +

= − +

= − +

= −

−=

−=



( ) ( )

( )

( ) ( ) ( )

( ) ( ) ( ) ( )

2 3 2

3 2

3 23 2

3 23 2

b 1 5 5

i For 0, 5 0 0 0 5 5

ii For 1, 5 1 1 1 5

1 1 1 5 8

iii For 1, 5 1 1 1 5

1 1 1 5 4

a a a a a a

a a a a

a a a a

a a a a

+ + + = + + +

= + + + = + + + =

= + + + = + + +

= + + + =

= − + + + = − + − + − +

= − + − + =

Q5. (a) Add: ( ) ( ) ( ), and p p q q q r r r p− − −

(b) Add: ( ) ( )2 2x z x y and y z y x− − − −

(c) Subtract: ( ) ( )3 4 5 from 4 10 3 2l l m n l n m l− + − +

(d) Subtract: ( ) ( ) ( )3 2 from 4a a b c b a b c c a b c+ + − − + − + +


Known:

Expressions

Unknown:

Simplification

Reasoning:

Addition and Subtraction takes place between like terms.

Solution:

(a) First expression = ( ) 2 p p q p pq− = −

Second expression = ( ) 2 q q r q qr− = −

Third expression = ( ) 2r r p r pr− = −

Adding the three expressions, we obtain 2

2

2

2 2 2

p pq

q qr

r pq

p pq q qr r pq

−

+ −

+ −

− + − + −

Therefore, the sum of the given expressions is 2 2 2 .p q r pq qr rp+ + − − −



(b) First expression = ( ) 22 2 2 2x z x y xz x xy− − = − −

Second expression = ( ) 22 2 2 2y z y x yz y yx− − = − −

Adding the two expressions, we obtain 2

2

2 2

2 2 2

2 2 2

2 2 4 2 2

xz x xy

xy yz y

xz x xy yz y

− −

+ − + −

− − + −

Therefore, the sum of the given expressions is 2 22 2 4 2 2 .x y xy yz zx− − − + +

( ) ( )

( )

2

2

c 3 4 5 3 12 15

4 10 3 2 40 12 8

l l m n l lm ln

l n m l ln lm l

− + = − +

− + = − +

Subtracting these expressions, we obtain

( ) ( ) ( )

2

2

2

40 12 8

15 12 3

25 5

ln lm l

l lm l

ln l

− +

− +

− + −

+ +

Therefore, the result is 25 25 .l ln+

( ) ( ) ( )

( )

2 2

2 2

2

3 2

3 3 3 2 2 2

3 2 3 2

4 4 4 4

d a a b c b a b c

a ab ac ba b bc

a b ab ac bc

c a b c ac bc c

+ + − − +

= + + − + −

= + + + −

− + + = − + +

Subtracting these expressions, we obtain

( ) ( ) ( ) ( ) ( )

2

2 2

2 2 2

4 4 4

3 2 3 2

7 6 4 3 2

ac bc c

ac bc a b ab

ac bc c a b ab

− + +

− + + +

− + − − −

− + + − − −

Therefore, the result is 2 2 23 2 4 6 7 .a b c ab bc ac− − + − + −





Q1. Multiply the binomials.

(i) ( ) ( )2 5 and 4 3x x+ −

(ii) ( ) ( )8 and 3 4y y− −

(iii) ( ) ( )2.5 0.5 and 2.5 0.5l m l m− +

(iv) ( ) ( )3 5a b and x+ +

(v) ( ) ( )2 22 3 and 3 2pq q pq q+ −

(vi) 2 2 2 23 23 and 4

4 3a b a b

+ −


Known:

Expressions

Unknown:

Multiplication

Reasoning:




Solution:

(i)

( ) ( ) ( ) ( )

( )

2

2

2 5 4 3 2 4 3 5 4 3

8 6 20 15

8 14 15 By adding like terms

x x x x x

x x x

x x

+ − = − + −

= − + −

= + −

(ii)

( ) ( ) ( ) ( )

( )

2

2

8 3 4 3 4 8 3 4

3 4 24 32

3 28 32 By adding like terms

y y y y y

y y y

y y

− − = − − −

= − − +

= − +



(iii)

( ) ( ) ( ) ( )2 2

2 2

2.5 0.5 2.5 0.5 2.5 2.5 0.5 0.5 2.5 0.5

6.25 1.25 1.25 0.25

6.25 0.25

l m l m l l m m l m

l lm lm m

l m

− + = + − +

= + − −

= −

(iv)

( ) ( ) ( ) ( )3 5 5 3 5

5 3 15

a b x a x b x

ax a bx b

+ + = + + +

= + + +

(v)

( ) ( ) ( ) ( )2 2 2 2 2

2 2 3 3 4

2 2 3 4

2 3 3 2 2 3 2 3 3 2

6 4 9 6

6 5 6

pq q pq q pq pq q q pq q

p q pq pq q

p q pq q

+ − = − + −

= − + −

= + −

(vi)

( )

2 2 2 2 2 2 2 2

2 2 2 2 2 2

22 2 2 2 2 2 2 2

4 2 2 2 2 4

4 2 2 4

3 2 3 83 4 3 4

4 3 4 3

3 8 84 3 4

4 3 3

3 3 8 84 3 4 3

4 4 3 3

3 2 12 8

3 10 8

a b a b a b a b

a a b b a b

a a a b b a b b

a b a b a b

a a b b

+ − = + −

= − + −

= − + −

= − + −

= + −

Q2. Find the product.

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

2 2

2 2

i 5 2 3

ii 7 7

iii

iv 2

x x

x y x y

a b a b

p q p q

− +

+ −

+ +

− +


Known:

Expressions

Unknown:

Simplification



Reasoning:




Solution:

(i)

( ) ( ) ( ) ( )2

2

5 2 3 5 3 2 3

15 5 6 2

15 2

x x x x x

x x x

x x

− + = + − +

= + − −

= − −

(ii)

( ) ( ) ( ) ( )2 2

2 2

7 7 7 7 7

7 49 7

7 48 7

x y x y x x y y x y

x xy xy y

x xy y

+ − = − + −

= − + −

= + −

(iii)

( ) ( ) ( ) ( )2 2 2 2 2

3 2 2 3

a b a b a a b b a b

a a b ab b

+ + = + + +

= + + +

(iv)

( ) ( ) ( ) ( )2 2 2 2

3 2 2 3

2 2 2

2 2

p q p q p p q q p q

p p q pq q

− + = + − +

= + − −

Q3. Simplify.

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

2

2 3

2 2

2 2

i 5 5 25

ii 5 3 5

iii

iv 2

v 2 2

vi

vii 1.5 4 1.5 4 3 4.5 12

vii

x x

a b

t s t s

a b c d a b c d ac bd

x y x y x y x y

x y x xy y

x y x y x y

− + +

+ + +

+ −

+ − + − + + +

+ + + + −

+ − +

− + + − +

( ) ( ) ( )i a b c a b c+ + + −


Known:

Expressions



Unknown:

Simplification

Solution:

(i)

( ) ( ) ( ) ( )2 2

3 2

3 2

5 5 25 5 5 5 25

5 5 25 25

5 5

x x x x x

x x x

x x x

− + + = + − + +

= + − − +

= + −

(ii)

( ) ( ) ( ) ( )2 3 2 3 3

2 3 2 3

2 3 2 3

5 3 5 3 5 3 5

3 5 15 5

3 5 20

a b a b b

a b a b

a b a b

+ + + = + + + +

= + + + +

= + + +

(iii)

( ) ( ) ( ) ( )2 2 2 2 2

3 2 2 3

t s t s t t s s t s

t st s t s

+ − = − + −

= − + −

(iv)

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2

2

2 2

2 2

4

a b c d a b c d ac bd

a c d b c d a c d b c d ac bd

ac ad bc bd ac ad bc bd ac bd

ac ac ac ad ad bc bc bd bd bd

ac

+ − + − + + +

= − + − + + − + + +

= − + − + + − − + +

= + + + − + − + − −

=

(v)

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

2 2 2 2

2 2 2 2

2 2

2 2

2 2 2

2 2 2 2

2 2 2 2

3 4

x y x y x y x y

x x y y x y x x y y x y

x xy xy y x xy xy y

x x y y xy xy xy xy

x y xy

+ + + + −

= + + + + − + −

= + + + + − + −

= + + − + + − +

= − +

(vi)

( ) ( )

( ) ( )

( ) ( )

2 2

2 2 2 2

3 2 2 2 2 3

3 3 2 2 2 2

3 3

x y x xy y

x x xy y y x xy y

x x y xy x y xy y

x y xy xy x y x y

x y

+ − +

= − + + − +

= − + + − +

= + + − + −

= +



(vii)

( ) ( )

( ) ( )

( ) ( ) ( )

2 2

2 2

2 2

1.5 4 1.5 4 3 4.5 12

1.5 1.5 4 3 4 1.5 4 3 4.5 12

2.25 6 4.5 6 16 12 4.5 12

2.25 6 6 4.5 4.5 16 12 12

2.25 16

x y x y x y

x x y y x y x y

x xy x xy y y x y

x xy xy x x y y y

x y

− + + − +

= + + − + + − +

= + + − − − − +

= + − + − − + −

= −

(viii)

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

2 2 2

2 2 2

2 2 2 2

a b c a b c

a a b c b a b c c a b c

a ab ac ab b bc ca bc c

a b c ab ab bc bc ca ca

a b c ab

+ + + −

= + − + + − + + −

= + − + + − + + −

= + − + + + − + −

= + − +





Q1. Use a suitable identity to get each of the following products.

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

( )

( ) ( ) ( )

2 2 2 2

i 3 3

ii 2 5 2 5

iii 2 7 2 7

iv

v 1.1m 0.4 1.1m 0.4

vi

vii 6 7 6 7

viii

ix

1 13 3

2 2

3 3

2 4 2 4

x 7 9 7 9

x x

y y

a a

a

a a

x y x

b a b

x x

a c a c

y

a b a b

− −

+

+ +

+ +

− −

− +

+ − +

− +

− + −

+

+

− −


Known:

Expressions

Unknown:

Simplification

Reasoning:




Solution:

The products will be as follows.

(i)

( ) ( ) ( )

( ) ( )( ) ( ) ( )

2

2 2 2 2 2

2

3 3 3

2 3 3 2

6 9

x x x

x x a b a ab b

x x

+ + = +

= + + + = + +

= + +



(ii)

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

2

2 2 2 2 2

2

2 5 2 5 2 5

2 2 2 5 5 2

4 20 25

y y y

y y a b a ab b

y y

+ + = +

= + + + = + +

=

+ +

(iii)

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

2

2 2 2 2 2

2

2 7 2 7 2 7

2 2 2 7 7 2

4 28 49

a a a

a a a b a ab b

a a

− − = −

= − + − = − +

=

− +

(iv)

( ) ( ) ( )

2

22 2 2 2

2

1 1 13 3 3

2 2 2

1 13 2 3 2

2 2

19 3

4

a a a

a a a b a ab b

a a

− − = −

= − + − = − +

= − +

(v)

( ) ( ) ( ) ( ) ( ) ( )2 2 2 2

2

1.1m 0.4 1.1m 0.4 1.1m 0.4

1.21m 0.16

a b a b a b − + = − +

= −

− = −

(vi)

( ) ( ) ( ) ( )

( ) ( ) ( )( )

2 2 2 2 2 2 2 2

2 22 2 2 2

4 4

a b a b b a b a

b a a b a b a b

b a

+ − + = + −

= − + − = −

= −

(vii)

( ) ( ) ( ) ( ) ( ) ( )2 2 2 2

2

6 7 6 7 6 7

36 49

x x x a b a b a b

x

− + = − + −

=

= −

−

(viii)

( ) ( ) ( )

( ) ( )( ) ( ) ( )

2

2 2 2 2 2

2 2

2 2

2

a c a c a c

a a c c a b a ab b

a ac c

− + − + = − +

= − + − + + =

−

=

+ +

+



(ix)

( )

2

2 22 2 2

2 2

3 3 3

2 4 2 4 2 4

3 32 2

2 2 4 4

3 9

4 4 16

x y x y x y

x x y ya b a ab b

x xy y

+ + = +

= + + + = + +

= + +

(x)

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

2

2 2 2 2 2

2 2

7 9 7 9 7 9

7 2 7 9 9 2

49 126 81

a b a b a b

a a b b a b a ab b

a ab b

− − = −

= − + − = − +

= − +

Q2. Use the identity ( ) ( ) ( )2x a x b x a b x ab+ + = + + + to find the following

products.

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

2 2

i 3 7

ii 4 5 4 1

iii 4 5 4 1

iv 4 5 4 1

v 2 5 2 3

vi 2 9 2 5

vii 4 2

x x

x x

x x

x x

x y x y

a a

xyz xyz

+ +

+ +

− −

+ −

+ +

+ +

− −


Known:

( )( ) ( )2 x a x b x a b x ab+ + = + + +

Unknown:

Simplification

Solution:

The products will be as follows.

(i)

( ) ( ) ( ) ( ) ( )2

2

3 7 3 7 3 7

10 21

x x x x

x x

+ + = + + +

= + +



(ii)

( ) ( ) ( ) ( ) ( ) ( ) ( )2

2

4 5 4 1 4 5 1 4 5 1

16 24 5

x x x x

x x

+ + = + + +

= + +

(iii)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2

2

4 5 4 1 4 5 1 4 5 1

16 24 5

x x x x

x x

− − = + − + − + − −

= − +

(iv)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2

2

4 5 4 1 4 5 1 4 5 1

16 16 5

x x x x

x x

+ − = + + + − + + −

= + −

(v)

( ) ( ) ( ) ( ) ( ) ( ) ( )2

2 2

2 5 2 3 2 5 3 2 5 3

4 16 15

x y x y x y y x y y

x xy y

+ + = + + +

= + +

(vi)

( ) ( ) ( ) ( ) ( ) ( ) ( )2

2 2 2 2

4 2

2 9 2 5 2 9 5 2 9 5

4 28 45

a a a a

a a

+ + = + + +

= + +

(vii)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2

2 2 2

4 2 4 2 4 2

6 8

xyz xyz xyz xyz

x y z xyz

− − = + − + − + − −

= − +

Q3. Find the following squares by using the identities.

( ) ( )

( ) ( )

( ) ( )

( )

( ) ( )

( ) ( )

2

2

2

2

2

2

2

2 3

3 2

i 7

ii 3

iii 6 5

iv

v 0.4 0.5

vi 2 5

b

xy z

x y

p

n

x

m

q

y y

−

+

−

−

+

+


Known:

Expressions



Unknown:

Simplification

Reasoning: 2 2 2

2 2 2

2 2

( ) 2

( ) 2

( )( )

a b a ab b

a b a ab b

a b a b a b

+ = + +

− = − +

+ − = −

Solution:

(i)

( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2

2

7 2 7 7 2

14 49

b b b a b a ab b

b b

− = − + − = − +

= − +

(ii)

( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2

2 2 2

3 2 3 3 2

6 9

xy z xy xy z z a b a ab b

x y xyz z

+ = + + + = + +

=

+ +

(iii)

( ) ( ) ( ) ( ) ( ) ( )2 2 2 22 2 2 2 2

4 2 2

6 5 6 2 6 5 5 2

36 60 25

x y x x y y a b a ab b

x x y y

− = − + − = − +

− +

=

(iv)

( )2 2 2

2 2

2 2

22 3 2 2 3 32

3 2 3 3 2 2

4 92

9 4

2m n m m n n

m m

a a a b

n n

b b + = + +

+

+

+

+

=

= +

(v)

( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2

2 2

0.4 0.5 0.4 2 0.4 0.5 0.5 2

0.16 0.4 0.25

p q p p q q a b a ab b

p pq q

− = − + − = − +

− +

=

(vi)

( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2

2 2 2 2

2 5 2 2 2 5 5 2

4 20 25

xy y xy xy y y a b a ab b

x y xy y

+ = + + + = + +

= +

+



Q4. Simplify.

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( )

22 2

2 2

2 2

2 2

2 2

2 2

22 2 3 2

i

ii 2 5 2 5

iii 7 8 7 8

iv 4 5 5 4

v 2.5 1.5 1.5 2.5

vi 2

vii 2

a b

x x

m n m n

m n m n

p q p q

ab bc ab c

m n m m n

−

+ − −

− + +

+ + +

− − −

+ −

− +


Known:

Expressions

Unknown:

Simplification

Reasoning: 2 2 2

2 2 2

2 2

( ) 2

( ) 2

( )( )

a b a ab b

a b a ab b

a b a b a b

+ = + +

− = − +

+ − = −

Solution:

( ) ( )

( ) ( )( ) ( ) ( )

22 2

2 2 22 2 2 2 2 2

4 2 2 4

2 2

2

i a b

a a b b a b a ab b

a a b b

−

= − + − = − +

= − +

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

( )

( )

2 2

2 2 2 2

2 2 2

2 2 2

2 2

2 2

2 5 2 5

2 2 2 5 5 2 2 2 5 5

2

2

4 20 25 4 20 25

4 20 25 4 20 25

40

ii x x

x x x x

a b a ab b

a b a ab b

x x x x

x x x x

x

+

+ − −

= + + − − +

− = − +

+ = +

= + + −

=

− +

+ + −

+ −

=



( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

2 2

2 2 2 2

2 22 2 2 2

2

7 8 7 8

7 2 7 8 8 7 2 7 8 8

2 2

49 112

iii m n m n

m m n n m m n n

a b a ab b and a b a ab b

m

−

− + +

= + + + +

− = − + + = + +

= − 2 2 64 49 112mn n m+ + + 2

2 2

64

98 128

mn n

m n

+

= +

( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )

2 2

2 2 2 2 2 2 2

2 2 2 2

2 2

4 5 5 4

4 2 4 5 5 5 2 5 4 4 2

16 40 25 25 40 16

41 80 41

iv m n m n

m m n n m m n n a b a ab b

m mn n m mn n

m mn n

+ + +

= + + + + + + = + +

= + + + + +

= + +

( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( )( ) ( )

( )

2 2

2 2 2 2

2 2 2

2 2 2 2

2

2.5 1.5 1.5 2.5

2.5 2 2.5 1.5 1.5 1.5 2 1.5 2.5 2.5

2

6.25 7.5 2.25 2.25 7.5 6.25

6.25 7.5

v p q p q

p p q q p p q q

a b a ab b

p pq q p pq q

p

− − −

= − + − − +

− = − +

= − + − − +

=

−

2.25pq + 2 2.25q − 2 7.5p + 2

2 24

] 6.25

4

pq q

p q

−

= −

( ) ( )

( ) ( )( ) ( ) ( )

2 2

2 2 22 2 2

2 2 2

2

2 2 2

2

vi ab bc ab c

ab ab bc bc ab c a b a ab b

a b ab c

+ −

= + + − + =

= +

+ +

2 2 2 2b c ab c+ −2 2 2 2 a b b c= +

( ) ( )

( ) ( ) ( ) ( ) ( )

22 2 3 2

2 2 22 2 2 2 3 2 2 2

4 3 2

2

2 2 2

2

vii m n m m n

m m n m n m m n a b a ab b

m m n

− +

= − + + − = − +

= − 4 2 3 22n m m n+ +4 4 2 m n m= +



Q5. Show that

( ) ( ) ( )

( ) ( ) ( )

( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

2 2

2 2

2 2 2

2

2 2

i 3 7 84 3 7

ii 9 5 180 9 5

ii4 3 16 9

23 4 9 1

i

iv 4 3 4 3 48

0

6

v

m n mn

x x x

p q pq p q

pq q pq q pq

a b a b b c b

m

c c a c a

n

+ − = −

− + = +

+ − − =

− + + − + + −

− + = +

+

=

Difficulty level: Hard

Known:

LHS and RHS expression

Unknown:

Verification of LHS = RHS

Solution:

( ) ( )

( ) ( ) ( ) ( )

( )

( ) ( ) ( ) ( )

2

2 2

2

2

2

2 2

2

i . . 3 7 84

3 2 3 7 7 84

9 42 49 84

9 42 49

. . 3 7

3 2 3 7 7

9 42 49

. . . .

L H S x x

x x x

x x x

x x

R H S x

x x

x x

L H S R H S

= + −

= + + −

= + + −

= − +

= −

= − +

= − +

=

( ) ( )

( ) ( ) ( ) ( )

( )

( ) ( ) ( ) ( )

2

2 2

2 2

2 2

2

2 2

2 2

ii . . 9 5 180

9 2 9 5 5 180

81 90 25 180

81 90 25

. . 9 5

9 2 9 5 5

81 90 25

. . . .

L H S p q pq

p p q q pq

p pq q pq

p pq q

R H S p q

p p q q

p pq q

L H S R H S

= − +

= − + +

= − + +

= + +

= +

= + +

= + +

=



( )2

2 2

2

4 3iii . . 2

3 4

4 4 3 32 2

3 3 4 4

162

9

L H S m n mn

m m n n mn

m mn

= − +

= − + +

= − 292

16n mn+ +

2 216 9

9 16

. . . .

m n

L H S R H S

= +

=

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2 2

2 2 2 2

2 2 2 2 2 2 2 2

2 2

iv . . 4 3 4 3

4 2 4 3 3 4 2 4 3 3

16 24 9 16 24 9

16

L H S pq q pq q

pq pq q q pq pq q q

p q pq q p q pq q

p q

= + − −

= + + − − +

= + + − − +

= 2 224 9pq q+ + 2 216 p q− 2 224 9pq q+ −

248

. . . .

pq

L H S R H S

=

=

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )2 2 2 2 2 2

2

v . . L H S a b a b b c b c c a c a

a b b c c a

a

= − + + − + + − +

= − + − + −

= 2b− 2b+ 2c− 2c+ 2a−

0

. . . . .L H S R H S

=

=

Q6.Using identities, evaluate.

( )

( )

( )

( )

( ) ( )

( )

( )

( )

( )

2

2

2

2

2

2

i 71

ii 99

iii 102

iv 998

v 5.2

vi 297 303

vii 78 82

viii 8.9

ix 1.05 9.5



Difficulty level: Hard

Known:

Expressions

Unknown:

Values of the expressions

Reasoning: 2 2 2

2 2 2

2 2

( ) 2

( ) 2

( )( )

a b a ab b

a b a ab b

a b a b a b

+ = + +

− = − +

+ − = −

Solution:

( ) ( )

( ) ( ) ( ) ( ) ( )

22

2 2 2 2 2

i 71 70 1

70 2 70 1 1 2

4900 140 1

5041

a b a ab b

= +

= + + + = + +

= + +

=

( )

( ) ( ) ( ) ( ) ( )

2 2

2 2 2 2 2

ii 99 1 00 1

100 2 100 1 1 2

10000 200 1

980

(

1

)

a b a ab b

= −

= − + − = − +

= − +

=

( ) ( )

( ) ( ) ( ) ( ) ( )

22

2 2 2 2 2

iii 102 100 2

100 2 100 2 2 2

10000 400 4

10404

a b a ab b

= +

= + + + =

=

+

=

+ +

+

( ) ( )

( ) ( ) ( ) ( ) ( )

22

2 2 2 2 2

iv 998 1000 2

1000 2 1000 2 2 2

1000000 4000 4

996004

a b a ab b

= −

= − + − = − +

= − +

=



( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

2 2

2 2 2 2 2

v 5.2 5.0 0.2

5.0 2 5.0 0.2 0.2 2

25 2 0.04

27.04

a b a ab b

= +

= + + + = + +

= + +

=

( ) ( ) ( )

( ) ( ) ( ) ( )2 2 2 2

vi 297 303 300 3 300 3

300 3

90000 9

89991

a b a b a b

= − +

= − + − = −

= −

=

( ) ( ) ( )

( ) ( ) ( ) ( )2 2 2 2

vii 78 82 80 2 80 2

80 2

6400 4

6396

a b a b a b

= − +

= − + − = −

= −

=

( ) ( )

( ) ( ) ( ) ( ) ( )

22

2 2 2 2 2

viii 8.9 9.0 0.1

9.0 2 9.0 0.1 0.1 2

81 1.8 0.01

79.21

a b a ab b

= −

= − + − = − +

= −

=

+

( )

( ) ( )

( ) ( )

( ) ( )

2 2

2 2

ix 1.05 9.5 1.05 0.95 10

1 0.05 1 0.05 10

1 0.05 10

1 0.0025 10

0.9975 10

9.975

a b a b a b

=

= + −

= −

= − + − =

=

=

−

Q 7. Using ( ) ( )2 2 ,a b a b a b− = + − find

( )

( ) ( ) ( )

( )

( )

2 2

2 2

2 2

2 2

i 51 – 49

ii 1.02 0.98

iii 153 147

iv 12.1 7.9

−

−

−




Known:

( ) ( )2 2 ,a b a b a b− = + −

Unknown:

Results of the given expression with their corresponding values

Solution:

( ) ( ) ( )

( ) ( )

2 2i 51 49 51 49 51 49

100 2 200

− = + −

= =

( ) ( ) ( ) ( ) ( )

( ) ( )

2 2ii 1.02 0.98 1.02 0.98 1.02 0.98

2 0.04

0.08

− = + −

=

=

( ) ( ) ( )

( ) ( )

2 2iii 153 147 153 147 153 147

300 6

1800

− = + −

=

=

( ) ( ) ( )

( ) ( )

2 2iv 12.1 7.9 12.1 7.9 12.1 7.9

20.0 4.2

84

− = + −

=

=

Q8. Using ( ) ( ) ( )2x a x b x a b x ab+ + = + + + , find

( )

( )

( )

( )

i 103 104

ii 5.1 5.2

iii 103 98

iv 9.7 9.8


Known:

( ) ( ) ( )2x a x b x a b x ab+ + = + + +

Unknown:

Results of the given expression with their corresponding values



Solution:

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )2

i 103 104 100 3 100 4

100 3 4 100 3 4

10000 700 12

10712

= + +

= + + +

= + +

=

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )2

ii 5.1 5.2 5 0.1 5 0.2

5 0.1 0.2 5 0.1 0.2

25 1.5 0.02

26.52

= + +

= + + +

= + +

=

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )2

iii 103 98 100 3 100 2

100 3 2 100 3 2

10000 100 6

10094

= + −

= + + − + −

= + −

=

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( )

2

iv 9.7 9.8 10 0.3 10 0.2

10 0.3 0.2 10 0.3 0.2

100 0.5 10 0.06

100 5 0.06

95 0.06

95.06

= − −

= + − + − + − −

= + − +

= − +

= +

=



chapter 9: algebraic expressions and identities

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