Page 1
Points to Remember :
Algebra. Algebra is generalised arithmetic in which numbers are represented by letters, known
as literal numbers or simply literals.
Since literals are also numbers, they obey all the rules of addition, subtraction, multiplication and
division.
Operations on Literals and Numbers :
1. Addition :
(i) The sum of literal x and a number 5 is x + 5
(ii) y more than x is written as x + y
(iii) For any literals a, b, c we have
a + b = b + a ; a + 0 = 0 + a = a and (a + b) + c = a + (b + c)
2. Subtraction :
(i) 5 less then a literal x is x – 5 ;
(ii) y less than x is x – y.
3. Multiplication :
(i) 4 times x is 4 × x, written as 4x.
(ii) The product of x and y is x × y, written as xy.
(iii) For any literals a, b, c we have :
a × 0 = 0 a = 0 ; a 1 = 1 a = a ; a b = b a ; (ab) c = a (bc) ; a (b + c)
= ab + ac.
4. Division :
(i) x divided by y is written as
x
y
(ii) x divided by 5 is
x
5
.
10 divided by x is
10
x
Powers of a literal
x x is written as x
2
, called x squared.
x x x is written as x
3
, called x cubed
x x x x is written as x
4
, called x raised to the power 4 and so on
In x
4
, x is called the base and 4 the exponent or index.
Page 2
Points to Remember :
Algebra. Algebra is generalised arithmetic in which numbers are represented by letters, known
as literal numbers or simply literals.
Since literals are also numbers, they obey all the rules of addition, subtraction, multiplication and
division.
Operations on Literals and Numbers :
1. Addition :
(i) The sum of literal x and a number 5 is x + 5
(ii) y more than x is written as x + y
(iii) For any literals a, b, c we have
a + b = b + a ; a + 0 = 0 + a = a and (a + b) + c = a + (b + c)
2. Subtraction :
(i) 5 less then a literal x is x – 5 ;
(ii) y less than x is x – y.
3. Multiplication :
(i) 4 times x is 4 × x, written as 4x.
(ii) The product of x and y is x × y, written as xy.
(iii) For any literals a, b, c we have :
a × 0 = 0 a = 0 ; a 1 = 1 a = a ; a b = b a ; (ab) c = a (bc) ; a (b + c)
= ab + ac.
4. Division :
(i) x divided by y is written as
x
y
(ii) x divided by 5 is
x
5
.
10 divided by x is
10
x
Powers of a literal
x x is written as x
2
, called x squared.
x x x is written as x
3
, called x cubed
x x x x is written as x
4
, called x raised to the power 4 and so on
In x
4
, x is called the base and 4 the exponent or index.
( ) EXERCISE 8 A
Q. 1. Write the following using literals,
numbers and signs of basic operations:
(i) x increased by 12
(ii) y decreased by 7
(iii) The difference of a and b, when a > b
(iv) The product of x and y added to their
sum
(v) One third of x multiplied by the sum of
a and b
(vi) 5 times x added to 7 times y
(vii) Sum of x and quotient of y by 5
(viii) x taken away from 4
(ix) 2 less than the quotient of x by y
(x) x multiplied by itself
(xi) Twice x increased by y
(xii) Thrice x added to y squared
(xiii) x minus twice y
(xiv) x cubed less than y cubed.
(xv) Quotient of x by 8 is multiplied by y.
Sol. (i) x + 12 (ii) y – 7
(iii) a – b (iv) (x + y) + xy
(v)
1
3
x a b ( ) (vi) 7y + 5x
(vii) x
y
5
(viii) 4 – x
(ix)
x
y
2
(x) x
2
(xi) 2x + y (xii) y
2
+ 3x
(xiii) x – 2y (xiv) y
3
– x
3
(xv)
x
y
8
Q. 2. Ranjit scores 80 marks in English and x
marks in Hindi. What is his total score
in the two subjects ?
Sol. Marks scored in English = 80
Marks scored in Hindi = x
Total score in the two subjects = 80 + x
Q. 3. Write the following in exponential form :
(i) b b b ......... 15 times
(ii) y y y ............. 20 times
(iii) 14 a a a a b b b
(iv) 6 x x y y
(v) 3 z z z y y x
Sol. We can write :
(i) b b b ................. 15 times = b
15
(ii) y y y .................. 20 times = y
20
(iii) 14 a a a a b b b = 14a
4
b
3
(iv) 6 x x y y = 6x
2
y
2
(v) 3 z z z y y x = 3z
3
y
2
x
Q. 4. Write down the following in product
form :
(i) x
2
y
4
(ii) 6y
5
(iii) 9xy
2
z (iv) 10a
3
b
3
c
3
Sol. We can write :
(i) x
2
y
4
= x x y y y y
(ii) 6y
5
= 6 y y y y y
(iii) 9xy
2
z = 9 x y y z
(iv) 10a
3
b
3
c
3
= 10 a a a b b b
c c c
Algebraic Expressions
Constant. A symbol having a fixed
numerical value is called a constant.
Variable. A symbol which takes on
various numerical values is called a
variable.
Algebraic Expression. A combination
of constants and variables, connected
by the symbols +, –, and is called
an algebraic expression.
The several parts of the expression
separated by the sign + or – are called
the ‘terms’ of the expression.
Various types of algebraic expressions
are :
(i) Monomials. An expression
containing only one term is called a
monomial.
Page 3
Points to Remember :
Algebra. Algebra is generalised arithmetic in which numbers are represented by letters, known
as literal numbers or simply literals.
Since literals are also numbers, they obey all the rules of addition, subtraction, multiplication and
division.
Operations on Literals and Numbers :
1. Addition :
(i) The sum of literal x and a number 5 is x + 5
(ii) y more than x is written as x + y
(iii) For any literals a, b, c we have
a + b = b + a ; a + 0 = 0 + a = a and (a + b) + c = a + (b + c)
2. Subtraction :
(i) 5 less then a literal x is x – 5 ;
(ii) y less than x is x – y.
3. Multiplication :
(i) 4 times x is 4 × x, written as 4x.
(ii) The product of x and y is x × y, written as xy.
(iii) For any literals a, b, c we have :
a × 0 = 0 a = 0 ; a 1 = 1 a = a ; a b = b a ; (ab) c = a (bc) ; a (b + c)
= ab + ac.
4. Division :
(i) x divided by y is written as
x
y
(ii) x divided by 5 is
x
5
.
10 divided by x is
10
x
Powers of a literal
x x is written as x
2
, called x squared.
x x x is written as x
3
, called x cubed
x x x x is written as x
4
, called x raised to the power 4 and so on
In x
4
, x is called the base and 4 the exponent or index.
( ) EXERCISE 8 A
Q. 1. Write the following using literals,
numbers and signs of basic operations:
(i) x increased by 12
(ii) y decreased by 7
(iii) The difference of a and b, when a > b
(iv) The product of x and y added to their
sum
(v) One third of x multiplied by the sum of
a and b
(vi) 5 times x added to 7 times y
(vii) Sum of x and quotient of y by 5
(viii) x taken away from 4
(ix) 2 less than the quotient of x by y
(x) x multiplied by itself
(xi) Twice x increased by y
(xii) Thrice x added to y squared
(xiii) x minus twice y
(xiv) x cubed less than y cubed.
(xv) Quotient of x by 8 is multiplied by y.
Sol. (i) x + 12 (ii) y – 7
(iii) a – b (iv) (x + y) + xy
(v)
1
3
x a b ( ) (vi) 7y + 5x
(vii) x
y
5
(viii) 4 – x
(ix)
x
y
2
(x) x
2
(xi) 2x + y (xii) y
2
+ 3x
(xiii) x – 2y (xiv) y
3
– x
3
(xv)
x
y
8
Q. 2. Ranjit scores 80 marks in English and x
marks in Hindi. What is his total score
in the two subjects ?
Sol. Marks scored in English = 80
Marks scored in Hindi = x
Total score in the two subjects = 80 + x
Q. 3. Write the following in exponential form :
(i) b b b ......... 15 times
(ii) y y y ............. 20 times
(iii) 14 a a a a b b b
(iv) 6 x x y y
(v) 3 z z z y y x
Sol. We can write :
(i) b b b ................. 15 times = b
15
(ii) y y y .................. 20 times = y
20
(iii) 14 a a a a b b b = 14a
4
b
3
(iv) 6 x x y y = 6x
2
y
2
(v) 3 z z z y y x = 3z
3
y
2
x
Q. 4. Write down the following in product
form :
(i) x
2
y
4
(ii) 6y
5
(iii) 9xy
2
z (iv) 10a
3
b
3
c
3
Sol. We can write :
(i) x
2
y
4
= x x y y y y
(ii) 6y
5
= 6 y y y y y
(iii) 9xy
2
z = 9 x y y z
(iv) 10a
3
b
3
c
3
= 10 a a a b b b
c c c
Algebraic Expressions
Constant. A symbol having a fixed
numerical value is called a constant.
Variable. A symbol which takes on
various numerical values is called a
variable.
Algebraic Expression. A combination
of constants and variables, connected
by the symbols +, –, and is called
an algebraic expression.
The several parts of the expression
separated by the sign + or – are called
the ‘terms’ of the expression.
Various types of algebraic expressions
are :
(i) Monomials. An expression
containing only one term is called a
monomial.
Various types of algebraic expressions
are :
(i) Monomials. An expression containing
only one term is called a monomial.
(ii) Binomials : An expression
containing two terms is called a
Binomial.
(iii) Trinomials : An expression
containing three terms is called a
trinomial.
(iv) Quadrinomials : An expression
containing four terms is called a
quadrinomial.
(v) Polynomials : An expression
containing two or more terms is known
as a polynomial.
Factors : When two or more numbers
and literals are multiplied then each one
of them is called a factor of the product.
Coefficients : In a product of numbers
and literals, any of the factors is called
the coefficient of the product of other
factors.
Constant Term : A term of the
expression having no literal factor is
called a constant term.
Like Terms : The terms having same
literal factors are called like or similar
terms.
Unlike Terms : The terms not having
same literal factors are called unlike or
dissimilar terms.
( ) EXERCISE 8 B
Q. 1. If a = 2, b = 3, find the value of :
(i) a + b (ii) a
2
+ ab
(iii) ab – a
2
(iv) 2a – 3b
(v) 5a
2
– 2ab (vi) a
3
– b
3
Sol. (i) Substituting a = 2 and b = 3 in the
given expression, we get :
a + b = 2 + 3 = 5
(ii) Substituting a = 2 and b = 3 in the given
expression, we get :
a
2
+ ab = (2)
2
+ 2 × 3
= 4 + 6 = 10
(iii) Substituting a = 2 and b = 3 in the given
expression, we get :
ab – a
2
= 2 × 3 – (2)
2
= 6 – 4 = 2
(iv) Substituting a = 2 and b = 3 in the given
expression, we get :
2a – 3b = 2 × 2 – 3 3 = 4 – 9 = – 5
(v) Substituting a = 2 and b = 3 in the given
expression, we get :
5a
2
– 2ab = 5 × (2)
2
– 2 × 2 × 3
= 5 × 4 – 4 3
= 20 – 12 = 8
(vi) Substituting a = 2 and b = 3 in the given
expression, we get :
a
3
– b
3
= (2)
3
– (3)
3
= 2 × 2 × 2 – 3
× 3 × 3
= 8 – 27 = – 19
Q. 2. If x = 1, y = 2 and z = 5, find the value
of :
(i) 3x – 2y + 4z (ii) x
2
+ y
2
+ z
2
(iii) 2x
2
– 3y
2
+ z
2
(iv) xy + yz – zx
(v) 2x
2
y – 5yz + xy
2
(vi) x
3
– y
3
– z
3
Sol. (i) Substituting x = 1, y = 2 and z = 5 in
the given expression, we get :
3x – 2y + 4z = 3 × 1 – 2 × 2 + 4 × 5
= 3 – 4 + 20 = 23 – 4 = 19
(ii) Substituting x = 1, y = 2 and z = 5 in
the given expression, we get :
x
2
+ y
2
+ z
2
= (1)
2
+ (2)
2
+ (5)
2
= 1 + 4 + 25 = 30
(iii) Substituting x = 1, y = 2 and z = 5 in the
given expression, we get :
2x
2
– 3y
2
+ z
2
= 2 (1)
2
– 3 (2)
2
+ (5)
2
= 2 1 – 3 4 + 25
= 2 – 12 + 25
= 27 – 12 = 15.
(iv) Substituting x = 1, y = 2 and z = 5 in the
given expression, we get :
Page 4
Points to Remember :
Algebra. Algebra is generalised arithmetic in which numbers are represented by letters, known
as literal numbers or simply literals.
Since literals are also numbers, they obey all the rules of addition, subtraction, multiplication and
division.
Operations on Literals and Numbers :
1. Addition :
(i) The sum of literal x and a number 5 is x + 5
(ii) y more than x is written as x + y
(iii) For any literals a, b, c we have
a + b = b + a ; a + 0 = 0 + a = a and (a + b) + c = a + (b + c)
2. Subtraction :
(i) 5 less then a literal x is x – 5 ;
(ii) y less than x is x – y.
3. Multiplication :
(i) 4 times x is 4 × x, written as 4x.
(ii) The product of x and y is x × y, written as xy.
(iii) For any literals a, b, c we have :
a × 0 = 0 a = 0 ; a 1 = 1 a = a ; a b = b a ; (ab) c = a (bc) ; a (b + c)
= ab + ac.
4. Division :
(i) x divided by y is written as
x
y
(ii) x divided by 5 is
x
5
.
10 divided by x is
10
x
Powers of a literal
x x is written as x
2
, called x squared.
x x x is written as x
3
, called x cubed
x x x x is written as x
4
, called x raised to the power 4 and so on
In x
4
, x is called the base and 4 the exponent or index.
( ) EXERCISE 8 A
Q. 1. Write the following using literals,
numbers and signs of basic operations:
(i) x increased by 12
(ii) y decreased by 7
(iii) The difference of a and b, when a > b
(iv) The product of x and y added to their
sum
(v) One third of x multiplied by the sum of
a and b
(vi) 5 times x added to 7 times y
(vii) Sum of x and quotient of y by 5
(viii) x taken away from 4
(ix) 2 less than the quotient of x by y
(x) x multiplied by itself
(xi) Twice x increased by y
(xii) Thrice x added to y squared
(xiii) x minus twice y
(xiv) x cubed less than y cubed.
(xv) Quotient of x by 8 is multiplied by y.
Sol. (i) x + 12 (ii) y – 7
(iii) a – b (iv) (x + y) + xy
(v)
1
3
x a b ( ) (vi) 7y + 5x
(vii) x
y
5
(viii) 4 – x
(ix)
x
y
2
(x) x
2
(xi) 2x + y (xii) y
2
+ 3x
(xiii) x – 2y (xiv) y
3
– x
3
(xv)
x
y
8
Q. 2. Ranjit scores 80 marks in English and x
marks in Hindi. What is his total score
in the two subjects ?
Sol. Marks scored in English = 80
Marks scored in Hindi = x
Total score in the two subjects = 80 + x
Q. 3. Write the following in exponential form :
(i) b b b ......... 15 times
(ii) y y y ............. 20 times
(iii) 14 a a a a b b b
(iv) 6 x x y y
(v) 3 z z z y y x
Sol. We can write :
(i) b b b ................. 15 times = b
15
(ii) y y y .................. 20 times = y
20
(iii) 14 a a a a b b b = 14a
4
b
3
(iv) 6 x x y y = 6x
2
y
2
(v) 3 z z z y y x = 3z
3
y
2
x
Q. 4. Write down the following in product
form :
(i) x
2
y
4
(ii) 6y
5
(iii) 9xy
2
z (iv) 10a
3
b
3
c
3
Sol. We can write :
(i) x
2
y
4
= x x y y y y
(ii) 6y
5
= 6 y y y y y
(iii) 9xy
2
z = 9 x y y z
(iv) 10a
3
b
3
c
3
= 10 a a a b b b
c c c
Algebraic Expressions
Constant. A symbol having a fixed
numerical value is called a constant.
Variable. A symbol which takes on
various numerical values is called a
variable.
Algebraic Expression. A combination
of constants and variables, connected
by the symbols +, –, and is called
an algebraic expression.
The several parts of the expression
separated by the sign + or – are called
the ‘terms’ of the expression.
Various types of algebraic expressions
are :
(i) Monomials. An expression
containing only one term is called a
monomial.
Various types of algebraic expressions
are :
(i) Monomials. An expression containing
only one term is called a monomial.
(ii) Binomials : An expression
containing two terms is called a
Binomial.
(iii) Trinomials : An expression
containing three terms is called a
trinomial.
(iv) Quadrinomials : An expression
containing four terms is called a
quadrinomial.
(v) Polynomials : An expression
containing two or more terms is known
as a polynomial.
Factors : When two or more numbers
and literals are multiplied then each one
of them is called a factor of the product.
Coefficients : In a product of numbers
and literals, any of the factors is called
the coefficient of the product of other
factors.
Constant Term : A term of the
expression having no literal factor is
called a constant term.
Like Terms : The terms having same
literal factors are called like or similar
terms.
Unlike Terms : The terms not having
same literal factors are called unlike or
dissimilar terms.
( ) EXERCISE 8 B
Q. 1. If a = 2, b = 3, find the value of :
(i) a + b (ii) a
2
+ ab
(iii) ab – a
2
(iv) 2a – 3b
(v) 5a
2
– 2ab (vi) a
3
– b
3
Sol. (i) Substituting a = 2 and b = 3 in the
given expression, we get :
a + b = 2 + 3 = 5
(ii) Substituting a = 2 and b = 3 in the given
expression, we get :
a
2
+ ab = (2)
2
+ 2 × 3
= 4 + 6 = 10
(iii) Substituting a = 2 and b = 3 in the given
expression, we get :
ab – a
2
= 2 × 3 – (2)
2
= 6 – 4 = 2
(iv) Substituting a = 2 and b = 3 in the given
expression, we get :
2a – 3b = 2 × 2 – 3 3 = 4 – 9 = – 5
(v) Substituting a = 2 and b = 3 in the given
expression, we get :
5a
2
– 2ab = 5 × (2)
2
– 2 × 2 × 3
= 5 × 4 – 4 3
= 20 – 12 = 8
(vi) Substituting a = 2 and b = 3 in the given
expression, we get :
a
3
– b
3
= (2)
3
– (3)
3
= 2 × 2 × 2 – 3
× 3 × 3
= 8 – 27 = – 19
Q. 2. If x = 1, y = 2 and z = 5, find the value
of :
(i) 3x – 2y + 4z (ii) x
2
+ y
2
+ z
2
(iii) 2x
2
– 3y
2
+ z
2
(iv) xy + yz – zx
(v) 2x
2
y – 5yz + xy
2
(vi) x
3
– y
3
– z
3
Sol. (i) Substituting x = 1, y = 2 and z = 5 in
the given expression, we get :
3x – 2y + 4z = 3 × 1 – 2 × 2 + 4 × 5
= 3 – 4 + 20 = 23 – 4 = 19
(ii) Substituting x = 1, y = 2 and z = 5 in
the given expression, we get :
x
2
+ y
2
+ z
2
= (1)
2
+ (2)
2
+ (5)
2
= 1 + 4 + 25 = 30
(iii) Substituting x = 1, y = 2 and z = 5 in the
given expression, we get :
2x
2
– 3y
2
+ z
2
= 2 (1)
2
– 3 (2)
2
+ (5)
2
= 2 1 – 3 4 + 25
= 2 – 12 + 25
= 27 – 12 = 15.
(iv) Substituting x = 1, y = 2 and z = 5 in the
given expression, we get :
xy + yz – zx = 1 2 + 2 5 – 5 1
= 2 + 10 – 5
= 12 – 5 = 7.
(v) Substituting x = 1, y = 2 and z = 5 in the
given expression, we get :
2x
2
y – 5yz + xy
2
= 2 (1)
2
2 – 5 2
5 + 1 (2)
2
= 2 1 2 –10 5 1 4
= 4 – 50 + 4
= 8 – 50 = – 42
(vi) Substitutng x = 1, y = 2 and z = 5 in the
given expression, we get :
x
3
– y
3
– z
3
= (1)
3
– (2)
3
– (5)
3
= (1 1 1) – (2 2 2) – (5 5 5)
= 1 – 8 – 125
= 1 – 133 = – 132
Q. 3. If p = – 2, q = – 1 and r = 3, find the
value of :
(i) p
2
+ q
2
– r
2
(ii) 2p
2
– q
2
+ 3r
2
(iii) p – q – r
(iv) p
3
+ q
3
+ r
3
+ 3pqr
(v) 3p
2
q
+ 5pq
2
+ 2pqr
(vi) p
4
+ q
4
– r
4
Sol. (i) Substituting p = – 2, q = – 1 and r =
3 in the given expression, we get :
p
2
+ q
2
– r
2
= (– 2)
2
+ (–1)
2
– (3)
2
= 4 + 1 – 9
= 5 – 9 = – 4
(ii) Substituting p = – 2, q = – 1 and r = 3 in
the given expression, we get :
2p
2
– q
2
+ 3r
2
= 2 × (– 2)
2
– (– 1)
2
+ 3
× (3)
2
= 2 × 4 – 1 + 3 × 9
= 8 – 1 + 27 = 34
(iii) Substituting p = – 2, q = – 1 and r = 3 in
the given expression, we get :
p – q – r = (– 2) – (– 1) – 3
= – 2 + 1 – 3 = – 4
(iv) Substituting p = – 2, q = – 1 and r = 3 in
the given expression, we get :
p
3
+ q
3
+ r
3
+ 3pqr = (– 2)
3
+ (– 1)
3
+ (3)
3
+ 3 (– 2) × (– 1) 3
= (– 8) + (– 1) + 27 + 18
= – 8 –1 + 27 + 18
= – 9 + 45 = 36
(v) Substituting p = – 2, q = – 1 and r = 3 in
the given expression, we get :
3p
2
q + 5pq
2
+ 2pqr = 3 (–2)
2
(– 1)
+ 5 × (– 2) × (– 1)
2
+ 2 (– 2) (– 1)
3
= 3 4 (– 1) + 5 (– 2) 1 + 12
= – 12 – 10 + 12 = – 10
(vi) Substituting p = – 2, q = – 1 and r = 3 in
the given expression, we get :
p
4
+ q
4
– r
4
= (– 2)
4
+ (– 1)
4
– (3)
4
= 16 + 1 – 81
= 17 – 81 = – 64
Q. 4. Write the coefficient of :
(i) x in 13x (ii) y in – 5y
(iii) a in 6ab (iv) z in – 7xz
(v) p in – 2pqr (vi) y
2
in 8xy
2
z
(vii) x
3
in x
3
(viii) x
2
in – x
2
Sol. (i) The coefficient of x in 13x is 13
(ii) The coefficient of y in – 5y is – 5
(iii) The coefficient of a in 6ab is 6b
(iv) The coefficient of z in – 7xz is – 7x
(v) The coefficient of p in – 2pqr is – 2qr
(vi) The coefficient of y
2
in 8xy
2
z is 8xz
(vii) The coefficient of x
3
in x
3
is 1
(viii) The coefficient of x
2
in – x
2
is –1
Q. 5. Write the numerical coefficient of :
(i) ab (ii) – 6bc
(iii) 7xyz (iv) – 2x
3
y
3
z
Sol. (i) The numerical coefficeint of ab is 1
(ii) The numerical coefficient of – 6bc is
– 6
(iii) The numerical coefficient of 7xyz is 7
Page 5
Points to Remember :
Algebra. Algebra is generalised arithmetic in which numbers are represented by letters, known
as literal numbers or simply literals.
Since literals are also numbers, they obey all the rules of addition, subtraction, multiplication and
division.
Operations on Literals and Numbers :
1. Addition :
(i) The sum of literal x and a number 5 is x + 5
(ii) y more than x is written as x + y
(iii) For any literals a, b, c we have
a + b = b + a ; a + 0 = 0 + a = a and (a + b) + c = a + (b + c)
2. Subtraction :
(i) 5 less then a literal x is x – 5 ;
(ii) y less than x is x – y.
3. Multiplication :
(i) 4 times x is 4 × x, written as 4x.
(ii) The product of x and y is x × y, written as xy.
(iii) For any literals a, b, c we have :
a × 0 = 0 a = 0 ; a 1 = 1 a = a ; a b = b a ; (ab) c = a (bc) ; a (b + c)
= ab + ac.
4. Division :
(i) x divided by y is written as
x
y
(ii) x divided by 5 is
x
5
.
10 divided by x is
10
x
Powers of a literal
x x is written as x
2
, called x squared.
x x x is written as x
3
, called x cubed
x x x x is written as x
4
, called x raised to the power 4 and so on
In x
4
, x is called the base and 4 the exponent or index.
( ) EXERCISE 8 A
Q. 1. Write the following using literals,
numbers and signs of basic operations:
(i) x increased by 12
(ii) y decreased by 7
(iii) The difference of a and b, when a > b
(iv) The product of x and y added to their
sum
(v) One third of x multiplied by the sum of
a and b
(vi) 5 times x added to 7 times y
(vii) Sum of x and quotient of y by 5
(viii) x taken away from 4
(ix) 2 less than the quotient of x by y
(x) x multiplied by itself
(xi) Twice x increased by y
(xii) Thrice x added to y squared
(xiii) x minus twice y
(xiv) x cubed less than y cubed.
(xv) Quotient of x by 8 is multiplied by y.
Sol. (i) x + 12 (ii) y – 7
(iii) a – b (iv) (x + y) + xy
(v)
1
3
x a b ( ) (vi) 7y + 5x
(vii) x
y
5
(viii) 4 – x
(ix)
x
y
2
(x) x
2
(xi) 2x + y (xii) y
2
+ 3x
(xiii) x – 2y (xiv) y
3
– x
3
(xv)
x
y
8
Q. 2. Ranjit scores 80 marks in English and x
marks in Hindi. What is his total score
in the two subjects ?
Sol. Marks scored in English = 80
Marks scored in Hindi = x
Total score in the two subjects = 80 + x
Q. 3. Write the following in exponential form :
(i) b b b ......... 15 times
(ii) y y y ............. 20 times
(iii) 14 a a a a b b b
(iv) 6 x x y y
(v) 3 z z z y y x
Sol. We can write :
(i) b b b ................. 15 times = b
15
(ii) y y y .................. 20 times = y
20
(iii) 14 a a a a b b b = 14a
4
b
3
(iv) 6 x x y y = 6x
2
y
2
(v) 3 z z z y y x = 3z
3
y
2
x
Q. 4. Write down the following in product
form :
(i) x
2
y
4
(ii) 6y
5
(iii) 9xy
2
z (iv) 10a
3
b
3
c
3
Sol. We can write :
(i) x
2
y
4
= x x y y y y
(ii) 6y
5
= 6 y y y y y
(iii) 9xy
2
z = 9 x y y z
(iv) 10a
3
b
3
c
3
= 10 a a a b b b
c c c
Algebraic Expressions
Constant. A symbol having a fixed
numerical value is called a constant.
Variable. A symbol which takes on
various numerical values is called a
variable.
Algebraic Expression. A combination
of constants and variables, connected
by the symbols +, –, and is called
an algebraic expression.
The several parts of the expression
separated by the sign + or – are called
the ‘terms’ of the expression.
Various types of algebraic expressions
are :
(i) Monomials. An expression
containing only one term is called a
monomial.
Various types of algebraic expressions
are :
(i) Monomials. An expression containing
only one term is called a monomial.
(ii) Binomials : An expression
containing two terms is called a
Binomial.
(iii) Trinomials : An expression
containing three terms is called a
trinomial.
(iv) Quadrinomials : An expression
containing four terms is called a
quadrinomial.
(v) Polynomials : An expression
containing two or more terms is known
as a polynomial.
Factors : When two or more numbers
and literals are multiplied then each one
of them is called a factor of the product.
Coefficients : In a product of numbers
and literals, any of the factors is called
the coefficient of the product of other
factors.
Constant Term : A term of the
expression having no literal factor is
called a constant term.
Like Terms : The terms having same
literal factors are called like or similar
terms.
Unlike Terms : The terms not having
same literal factors are called unlike or
dissimilar terms.
( ) EXERCISE 8 B
Q. 1. If a = 2, b = 3, find the value of :
(i) a + b (ii) a
2
+ ab
(iii) ab – a
2
(iv) 2a – 3b
(v) 5a
2
– 2ab (vi) a
3
– b
3
Sol. (i) Substituting a = 2 and b = 3 in the
given expression, we get :
a + b = 2 + 3 = 5
(ii) Substituting a = 2 and b = 3 in the given
expression, we get :
a
2
+ ab = (2)
2
+ 2 × 3
= 4 + 6 = 10
(iii) Substituting a = 2 and b = 3 in the given
expression, we get :
ab – a
2
= 2 × 3 – (2)
2
= 6 – 4 = 2
(iv) Substituting a = 2 and b = 3 in the given
expression, we get :
2a – 3b = 2 × 2 – 3 3 = 4 – 9 = – 5
(v) Substituting a = 2 and b = 3 in the given
expression, we get :
5a
2
– 2ab = 5 × (2)
2
– 2 × 2 × 3
= 5 × 4 – 4 3
= 20 – 12 = 8
(vi) Substituting a = 2 and b = 3 in the given
expression, we get :
a
3
– b
3
= (2)
3
– (3)
3
= 2 × 2 × 2 – 3
× 3 × 3
= 8 – 27 = – 19
Q. 2. If x = 1, y = 2 and z = 5, find the value
of :
(i) 3x – 2y + 4z (ii) x
2
+ y
2
+ z
2
(iii) 2x
2
– 3y
2
+ z
2
(iv) xy + yz – zx
(v) 2x
2
y – 5yz + xy
2
(vi) x
3
– y
3
– z
3
Sol. (i) Substituting x = 1, y = 2 and z = 5 in
the given expression, we get :
3x – 2y + 4z = 3 × 1 – 2 × 2 + 4 × 5
= 3 – 4 + 20 = 23 – 4 = 19
(ii) Substituting x = 1, y = 2 and z = 5 in
the given expression, we get :
x
2
+ y
2
+ z
2
= (1)
2
+ (2)
2
+ (5)
2
= 1 + 4 + 25 = 30
(iii) Substituting x = 1, y = 2 and z = 5 in the
given expression, we get :
2x
2
– 3y
2
+ z
2
= 2 (1)
2
– 3 (2)
2
+ (5)
2
= 2 1 – 3 4 + 25
= 2 – 12 + 25
= 27 – 12 = 15.
(iv) Substituting x = 1, y = 2 and z = 5 in the
given expression, we get :
xy + yz – zx = 1 2 + 2 5 – 5 1
= 2 + 10 – 5
= 12 – 5 = 7.
(v) Substituting x = 1, y = 2 and z = 5 in the
given expression, we get :
2x
2
y – 5yz + xy
2
= 2 (1)
2
2 – 5 2
5 + 1 (2)
2
= 2 1 2 –10 5 1 4
= 4 – 50 + 4
= 8 – 50 = – 42
(vi) Substitutng x = 1, y = 2 and z = 5 in the
given expression, we get :
x
3
– y
3
– z
3
= (1)
3
– (2)
3
– (5)
3
= (1 1 1) – (2 2 2) – (5 5 5)
= 1 – 8 – 125
= 1 – 133 = – 132
Q. 3. If p = – 2, q = – 1 and r = 3, find the
value of :
(i) p
2
+ q
2
– r
2
(ii) 2p
2
– q
2
+ 3r
2
(iii) p – q – r
(iv) p
3
+ q
3
+ r
3
+ 3pqr
(v) 3p
2
q
+ 5pq
2
+ 2pqr
(vi) p
4
+ q
4
– r
4
Sol. (i) Substituting p = – 2, q = – 1 and r =
3 in the given expression, we get :
p
2
+ q
2
– r
2
= (– 2)
2
+ (–1)
2
– (3)
2
= 4 + 1 – 9
= 5 – 9 = – 4
(ii) Substituting p = – 2, q = – 1 and r = 3 in
the given expression, we get :
2p
2
– q
2
+ 3r
2
= 2 × (– 2)
2
– (– 1)
2
+ 3
× (3)
2
= 2 × 4 – 1 + 3 × 9
= 8 – 1 + 27 = 34
(iii) Substituting p = – 2, q = – 1 and r = 3 in
the given expression, we get :
p – q – r = (– 2) – (– 1) – 3
= – 2 + 1 – 3 = – 4
(iv) Substituting p = – 2, q = – 1 and r = 3 in
the given expression, we get :
p
3
+ q
3
+ r
3
+ 3pqr = (– 2)
3
+ (– 1)
3
+ (3)
3
+ 3 (– 2) × (– 1) 3
= (– 8) + (– 1) + 27 + 18
= – 8 –1 + 27 + 18
= – 9 + 45 = 36
(v) Substituting p = – 2, q = – 1 and r = 3 in
the given expression, we get :
3p
2
q + 5pq
2
+ 2pqr = 3 (–2)
2
(– 1)
+ 5 × (– 2) × (– 1)
2
+ 2 (– 2) (– 1)
3
= 3 4 (– 1) + 5 (– 2) 1 + 12
= – 12 – 10 + 12 = – 10
(vi) Substituting p = – 2, q = – 1 and r = 3 in
the given expression, we get :
p
4
+ q
4
– r
4
= (– 2)
4
+ (– 1)
4
– (3)
4
= 16 + 1 – 81
= 17 – 81 = – 64
Q. 4. Write the coefficient of :
(i) x in 13x (ii) y in – 5y
(iii) a in 6ab (iv) z in – 7xz
(v) p in – 2pqr (vi) y
2
in 8xy
2
z
(vii) x
3
in x
3
(viii) x
2
in – x
2
Sol. (i) The coefficient of x in 13x is 13
(ii) The coefficient of y in – 5y is – 5
(iii) The coefficient of a in 6ab is 6b
(iv) The coefficient of z in – 7xz is – 7x
(v) The coefficient of p in – 2pqr is – 2qr
(vi) The coefficient of y
2
in 8xy
2
z is 8xz
(vii) The coefficient of x
3
in x
3
is 1
(viii) The coefficient of x
2
in – x
2
is –1
Q. 5. Write the numerical coefficient of :
(i) ab (ii) – 6bc
(iii) 7xyz (iv) – 2x
3
y
3
z
Sol. (i) The numerical coefficeint of ab is 1
(ii) The numerical coefficient of – 6bc is
– 6
(iii) The numerical coefficient of 7xyz is 7
(iv) The numerical coefficient of – 2x
3
y
3
z is
– 2.
Q. 6. Write the constant term of :
(i) 3x
2
+ 5x + 8
(ii) 2x
2
– 9
(iii) 4 5
3
5
2
y y
(iv)
z z z
3 2
2
8
3
Sol. (i) The constant term is 8
(ii) The constant term is – 9
(iii) The constant term is
3
5
(iv) The constant term is
8
3
Q. 7. Identify the monomials, binomials and
trinomials in the following :
(i) – 2xyz (ii) 5 + 7x
3
y
3
z
3
(iii) – 5x
3
(iv) a + b – 2c
(v) xy + yz – zx (vi) x
5
(vii) ax
3
+ bx
2
+ cx + d
(viii) – 14
(ix) 2x + 1
Sol. (i) The given expression contains only
one term, so it is monimial.
(ii) The given expression contains only two
terms, so it is binomial.
(iii) The given expression contains only one
term, so it is monomial.
(iv) The given expression contains three
terms, so it is trinomial.
(v) The given expression contains three
terms, so it is trinomial.
(vi) The given expression contains only one
term, so it is monomial.
(vii) The given expression contains four
terms, so it is none of monomial,
binomial and trinomial.
(viii) The given expression contains only one
term so it is monomial.
(ix) The given expression contains two
terms, so it is binomial.
Q. 8. Write all the terms of the algebraic
expressions :
(i) 4x
5
– 6y
4
+ 7x
2
y – 9
(ii) 9x
3
– 5z
4
+ 7x
3
y – xyz
Sol. (i) The terms of the given expression
4x
5
– 6y
4
+ 7x
2
y – 9 are :
4x
5
, – 6y
4
, 7x
3
y, – 9
(ii) The terms of the given expression
9x
3
– 5z
4
+ 7x
3
y
– xyz are :
9x
3
, – 5z
4
, 7x
3
y, – xyz.
Q. 9. Identify like terms in the following :
(i) a
2
, b
2
, – 2a
2
, c
2
, 4a
(ii) 3x, 4xy, – yz,
1
2
zy
(iii) – 2xy
2
, x
2
y, 5y
2
x, x
2
z
(iv) abc, ab
2
c, acb
2
, c
2
ab, b
2
ac, a
2
bc, cab
2
Sol. (i) We have : a
2
, b
2
,
– 2a
2
, c
2
, 4a
Here like terms are a
2
, – 2a
2
(ii) We have : 3x, 4xy, –yz,
1
2
zy
Here like terms are yz zy ,
1
2
(iii) We have : – 2xy
2
, x
2
y, 5y
2
x, x
2
z
Here like terms are –2xy
2
, 5y
2
x
(iv) We have :
abc, ab
2
c, acb
2
, c
2
ab, b
2
ac, a
2
bc, cab
2
Here like terms are ab
2
c, acb
2
, b
2
ac,
cab
2
.
Operations on Algebraic Expressions
1. Addition of Algebraic Expressions
The sum of several like terms is another
like term whose coefficient is the sum
of the coefficients of the like terms.
Column Method. In this method, each
expression is written in a separate row
such that their like terms are arranged
one below the other in a column. Then
addition or subtraction of the terms is
done columnwise.
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