Unit-5(Algebra)

CHAPTER-9

TOPIC: CONSTANT AND VARIABLES

9.1Constant and Variables:

1. Constant: If a letter represents a fixed value (number), it is called constant.
(i) For example; x represents the number of provinces of Nepal.
(ii) = . So, x is a constant.
(iii) Constant is represented by English alphabets.
1. Variable: If a letter represents many values (numbers), it is called variable.
(i) For example; y represents the whole numbers between 1 and 5.
(ii) = , , . So, y is a variable.
(iii) Variables are also represented by English alphabets.

(Obj.1) To define and identify constant and variable. (4)

1. Define a constant. (1)
2. If x represents the natural numbers between 3 and 5, what does x represent? (1)
3. Define a variable. (1)
4. If y represents the heights of the students in your class, what does y represent? (1)

Solutions:

(Obj.1)

1. Solution: Here, if a letter represents a fixed value (number), it is called constant. (1)
2. Solution: Here, if x represents the natural numbers between 3 and 5, x represents a constant.

(1)
3. Solution: Here, if a letter represents many values (numbers), it is called variable. (1)
4. Solution: Here, if y represents the heights of the students in your class, y represents a variable.

(1)

9.2Algebraic terms and expressions:

1. Constant term: A constant term is a term in an algebraic expression that does not contain any
variables.

(i) For example; , + − , + − −

2. Algebraic term: An algebraic term is a group of constant and variable linked by × ÷ signs.
3. Types of algebraic terms: There are four types of algebraic terms. Those are as follows;

(i) Monomial term
(ii) Binomial term
(iii) Trinomial term
(iv) Multinomial term
4. Monomial term: An algebraic term which contains only one term is called monomial term.
(i) For example: , ,
5. Binomial term: An algebraic term which contains only two unlike terms is called binomial
term.
(i) For example: + , − , −
6. Trinomial term: An algebraic term which contains only unlike three terms is called trinomial
term.
(i) For example: + + , + −
7. Multinomial term: An algebraic term which contains two or more than two unlike terms is
called multinomial term.

(i) For example: + , + + , + − −
8. Algebraic expression: An algebraic expression is a group of algebraic terms separated by

+ − signs.
(i) For example: , , + , + − + − −
9. Types of algebraic expressions: There are four types of algebraic expressions. Those are as
follows:
(i) Monomial algebraic expression
(ii) Binomial algebraic expression
(iii) Trinomial algebraic expression
(iv) Multinomial algebraic expression
10. Monomial algebraic expression: An algebraic expression containing only one term is called a
monomial algebraic expression.
(i) For example; , ,
11. Binomial algebraic expression: An algebraic expression containing two unlike terms is called a
binomial algebraic expression.
(i) For example: + , − , +
12. Trinomial algebraic expression: An algebraic expression containing three unlike terms is
called a trinomial algebraic expression.
(i) For example: − + , + − , − +
13. Multinomial algebraic expression: An algebraic expression containing two or more than two
unlike terms is called a multinomial expression.
(i) For example: + + + , − + − , + − −
14. Polynomial: An algebraic expression having non-negative integer as the power of variables in
each term is called a polynomial.
(i) For example: , + , − + , + +
15. Mathematical statement: Mathematical statement is a sentence which may be either true or
false but not both.
(i) For example: the sum of .

(Obj.2) To define a constant term.

1. Define a constant term. (1)

(Obj.3) To define an algebraic term.

1. Define an algebraic term. (1)

(Obj.4) To write the types of algebraic terms.

1. Write the types of algebraic terms. (4)

(Obj.5) To define the types of algebraic terms.

1. Define a monomial term. (1)
2. Define a multinomial term. (1)

(Obj.6) To define an algebraic expression.

1. Define an algebraic expression. (1)

(Obj.7) To write the types of algebraic expressions.

1. Write the types of algebraic expressions. (4)

(Obj.8) To define and identify the types of algebraic expressions.

1. Define binomial algebraic expression. (1)

2. State the type of: + + + + (1)

(Obj.9) To define a polynomial.

1. Define a polynomial. (1)

(Obj.10) To define a mathematical statement.

1. Define a mathematical statement. (1)

(Obj.11) To write the mathematical statement into algebraic expression.

1. Express the following statements into algebraic expressions.
(i) The sum of x and y is added to z. (1)
(ii) The product of p and q is subtracted from r. (1)
(iii) The perimeter of a triangle is sum of its three sides a, b and c of the triangle. What is the
formula of perimeter of the triangle? (1)
(iv) The area of a square is the square of its length ( ). What is the formula of area of the
square? (1)
(v) The perimeter of a rectangle is two times the sum of its length ( ) and breadth ( ).
What is the formula of perimeter of the rectangle? (1)

Solutions:

(Obj.2) To define a constant term.

1. Define a constant term. (1)
Solution: Here, A constant term is a term in an algebraic expression that does not contain any
variables.

(Obj.3) To define an algebraic term.

1. Define an algebraic term. (1)
Solution: Here, an algebraic term is a group of constant and variable linked by × ÷ signs.

(Obj.4) To write the types of algebraic terms.

1. Write the types of algebraic terms. (4)
Solution: Here, the types of algebraic terms are as follows;
(i) Monomial term
(ii) Binomial term
(iii) Trinomial term
(iv) Multinomial term

(Obj.5) To define the types of algebraic terms.

1. Define a monomial term. (1)

Solution: Here, an algebraic term which contains only one term is called monomial term.

2. Define a multinomial term. (1)

Solution: Here, an algebraic term which contains two or more than two unlike terms is called
multinomial term.

(Obj.6) To define an algebraic expression.

1. Define an algebraic expression. (1)

Solution: Here, an algebraic expression is a group of algebraic terms separated by + −
signs.

(Obj.7) To write the types of algebraic expressions.

1. Write the types of algebraic expressions. (4)
Solution: Here, the types of algebraic expressions are as follows;
(i) Monomial algebraic expression
(ii) Binomial algebraic expression
(iii) Trinomial algebraic expression
(iv) Multinomial algebraic expression

(Obj.8) To define and identify the types of algebraic expressions.

1. Define binomial algebraic expression. (1)

Solution: Here, an algebraic expression containing two unlike terms is called a binomial
algebraic expression.
2. State the type of: + + + + (1)
Solution: Here, + + + + is a multinomial algebraic expression.

(Obj.9) To define a polynomial.

1. Define a polynomial. (1)

Solution: Here, an algebraic expression having non-negative integer as the power of variables
in each term is called a polynomial.

(Obj.10) To define a mathematical statement.

1. Define a mathematical statement. (1)

Solution: Here, mathematical statement is a sentence which may be either true or false but not
both.

(Obj.11) To write the mathematical statement into algebraic expression.

2. Express the following statements into algebraic expressions.
(i) The product of x and y is added to z. (1)
Solution: Here, × + is the required algebraic expression.

(ii) The product of p and q is subtracted from r. (1)
Solution: Here, − × is the required expression.

(iii) The perimeter of a triangle is sum of its three sides a, b and c of the triangle. What is the
formula of perimeter of the triangle? (1)

Solution: Here, the formula of perimeter of the triangle (P) is + + .
(iv) The area of a square is the square of its length ( ). What is the formula of area of the

square? (1)
Solution: Here, the formula of area of the square (A) is .
(v) The perimeter of a rectangle is two times the sum of its length ( ) and breadth ( ).

What is the formula of perimeter of the rectangle? (1)
Solution: Here, the formula of perimeter of the rectangle (P) is ( + ).

9.3 Co-efficient, base and power/exponent/index of algebraic term:

1. Co-efficient: A co-efficient is a number multiplied by a variable.
(i) For example: 3 is a co-efficient of in .

2. Numerical co-efficient: A numerical co-efficient is a constant multiplier of the variables in a
term.
(i) For example: 4 is a numerical co-efficient of in .

3. Literal co-efficient: A numerical co-efficient is a variable multiplier of the variables in a term.
(i) For example: a is a literal co-efficient of in .

4. Base: The base is used in algebra in connection with powers.

The base is used in algebra in connection with powers. In fact, it is called the base of a
power—or the number that is used as a factor a given number of times. In the

example above, , 3 is the base. The base can either be the number used with an exponent to
create a power, such as the 3 in ; or a number written as a subscript, such as with a
logarithm, for example, , in which a is the base number.
(i) For example: 5 is the base in ^ .
5. Exponent: The exponent is the number that tells how many times the base is used as factor.
It is also called power or index.

➢ A power is a product of repeated multiplication of the same base.
➢ The exponent of a power is the number of times the base is multiplied.
(i) For example: 6 is the exponent of x in .

(Obj.12) To write the numerical co-efficient, literal co-efficient, base and power in a term.

1. What is the numerical co-efficient of in ? (1)
2. What is the literal co-efficient of in ? (1)
3. What is the base in ? (1)
4. What is the power of in ? (1)

Solutions:

1. What is the numerical co-efficient of in ? (1)

Solution: Here, the numerical co-efficient of in is 5.

2. What is the literal co-efficient of in ? (1)

Solution: Here, the literal co-efficient of in is x.

3. What is the base in ? (1)

Solution: Here, the base in is x.

4. What is the power of x in ? (1)

Solution: Here, the power of x is 5 in .

Note:

(i) Sum/added/increased holds + sign.
(ii) Subtracted/difference/decreased holds – sign.
(iii) Product/multiplied/times holds × sign.
(iv) Quotient/divided holds ÷ sign.

EXERCISE-9.1

CREATIVE SECTION

Q.5 Let’s take the terms x, y and z. Make monomial, binomial and trinomial expressions of your
own using these terms.
Solution: Here, the given terms are x, y and z.
Now,
(i) Monomial expression:
(ii) Binomial expression:
(iii) Trinomial expression:

Q.6 Rewrite the following statements in algebraic expressions.
a) Product of x and y is added to z
Solution: Here, × + is the required expression.
b) Three times the sum of x and y is increased by 5.
Solution: Here, ( + ) + is the required expression.
c) Two times the difference of p and q is decreased by 4.
Solution: Here, ( − ) − is the required expression.
d) Product of p and q is subtracted from r.
Solution: Here, − × is the required expression.
e) Five times the product of a and b is increased by x.
Solution: Here, × × + is the required expression.
f) The sum of x and y is divided by 2 and decreased by 7.
Solution: Here, ( + ) ÷ − is the required expression.

Q.7 a) The present age of is x years.
(i) How old was he 2 years before?
Solution: Here, he was ( − ) years old 2 years before.
(ii) How old will he be after 2 years?
Solution: Here, he will be ( + ) years old after 2 years.
(iii) If his father is four times older than him, how old is his father?

Solution: Here, if his father is four times older than him, then his father is years old.
b) The breadth of a room is b . If its length is 5 longer than its breadth,
represent the length of the room by an expression.
Solution: Here, the breadth of a room = m
The length of the room = ( + ) m
Hence, the required expression of the length of the room is ( + ) m.
c) The marks obtained by A in is x. The marks obtained by B is double than that of A
and marks obtained by C is double than that of B. Represent the marks obtained by B and C
by expressions.

Solution: Here, the marks obtained by in =

The marks obtained by B in =

The marks obtained by C in =

Hence, the required expressions for the marks obtained by B and C in are and
respectively.

Q.8 Rewrite the following formulae in algebraic expressions.

a) The perimeter of a triangle is sum of its three sides a, b and c of the triangle. What is the
formula of perimeter of the triangle?

Solution: Here, the formula of perimeter of the triangle (P) is ( + + ).

b) The area of a triangle is half of the product of base (b) and height (h). What is the formula
of area of the triangle?

Solution: Here, the formula of area of the triangle (A) is × × .

c) The perimeter of a square is four times its length ( ). What is the formula of perimeter of the
square?

Solution: Here, the formula of perimeter of the square (P) is .

d) The area of a square is the square of its length ( ). What is the formula of area of the
square?

Solution: Here, the formula of area of the square (A) is .

e) The perimeter of a rectangle is two times the sum of its length ( ) and breadth ( ). What is
the formula of perimeter of the rectangle?

Solution: Here, the formula of perimeter of the rectangle (P) is ( + ).

f) The area of a rectangle is the product of its length ( ) and breadth ( ). What is the formula
of area of the rectangle?

Solution: Here, the formula of area of the rectangle (A) is × .

It’s your time-Project work!

Q.9 Let’s write any two different monomial expressions. Write the co-efficient, base and exponent
of each expression.

Expressions Co-efficient Base Exponent

“The-End”