3. Chapter 3. Algebraic Expressions and Polynomials NIOS Class 10 Mathematics Textbook Solutions
Chapter 3. Algebraic Expressions and Polynomials NIOS Class 10 Mathematics Textbook Solutions for Exam Preparation
Chapter 3. ALGEBRAIC EXPRESSIONS AND POLYNOMIALS 
CHECK YOUR PROGRESS 3.1 ALGEBRAIC EXPRESSIONS AND POLYNOMIALS
1. Write the variables and constants in each of the following:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
Solution: (i)
The variable is y .
The constant is 1 .
(ii)
The variable are x and y .
The constants are 
(iii)
The variable are x and y .
The constant is 
.
(iv)
The variable are x and y .
The constants are 
.
(v)
The variable are x and y .
The constants are 2 and – 8 .
(vi)
The variable is x .
There are no constants .
2. In 
, write the coefficient of (i) 
(ii) 
(iii) 
Solution: (i) The coefficient of is 2 .
(ii) The coefficient of is 2y
(iii) The coefficient of 
is 
.
3. Using variables and operation symbols, express each of the following verbal statements as algebraic statements:
(i) three less than a number equals fifteen.
(ii) A number increased by five gives twenty-two.
Solution: (i) Let , x be the number .
A/Q, 
(ii) Let , x be the number .
A/Q, 
4. Write the terms of each of the following expressions:
(i) 
(ii) 
(iii) 
(iv) 
Solution: (i)
The terms are 2 and abc
(ii) 
The terms are a , b, c and 2 .
(iii) 
The terms are 
(iv)
The term is 
5. Identify like terms, if any, in each of the following expressions:
(i) 
(ii) 
(iii) 
Solution: (i)
Like terms are : 
(ii) 
Like terms are : 
(iii)
There are no like terms .
6. Which of the following algebraic expressions are polynomials?
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
Solution: (i) is polynomial .
(ii) is polynomial .
(iii) is not polynomial.
(iv) is not polynomial .
(v) is not polynomial.
(vi) is not polynomial .
7. Identify each of the following as a monomial, binomial or a trinomial:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
Solution: Monomial : 
and .
Binomial : and .
Trinomial: and
CHECK YOUR PROGRESS 3.2 ALGEBRAIC EXPRESSIONS AND POLYNOMIALS
1. Write the degree of each of the following monomials:
(i) 
(ii) 
(iii) 
(iv) 27
Solution: (i) The degree of the monomial of 
is 7 .
(ii) The degree of the monomial of 
is 3 .
(iii) The degree of the monomial of 
is 1 .
(iv) The degree of the monomial of 
is 0 .
2. Rewrite the following monomials in increasing order of their degrees:
Solution: We have, 
3. Determine the degree of each of the following polynomials:
(i) 
(ii) 
(iii) 
(iv) 
Solution: (i) The degree of the polynomials of is 10 (= 6 + 4) .
(ii) The degree of the polynomials of is 4 (= 1+3) .
(iii) The degree of the polynomials of is 2 .
(iv) The degree of the polynomials of is 3 (= 2+1=1+2)
[ Note: The sum of the exponents of the variables in a term is called the degree of that term.]
4. Evaluate each of the following polynomials for the indicated value of the variable:
(i) 
(ii) 
(iii) 
(iv) 
Solution: (i)
For 
, the value of the given polynomial 
(ii)
For , the value of the given polynomial 
(iii) 
For
, the value of the given polynomial
(iv)
For 
, the value of the given polynomial 
5. Verify that each of 
and 
is a zero of the polynomial 
.
Solution: For x = 2 :
For x = 3 :
Therefore, and are the zeros of the polynomial .
CHECK YOUR PROGRESS 3.3 ALGEBRAIC EXPRESSIONS AND POLYNOMIALS
1. Add the following pairs of polynomials:
(i) 
(ii) 
(iii) 
(iv) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
(iv) We have, 
2. Add : (i) 
(ii) 
(iii) 
(iv) 
Solution: (i) 
(ii) We have, 
(iii) We have, 
(iv) We have, 
3. Subtract : (i) 
(ii) 
(iii) 
(iv) 
Solution: (i) We have, 
(ii) we have, 
(iii) We have,
(iv) We have, 
4. Subtract 
from the sum of 
and 
.
Solution: We have, 
Now , 
CHECK YOUR PROGRESS 3.4 ALGEBRAIC EXPRESSIONS AND POLYNOMIALS
1. Multiply :
(i) 
(ii) 
(iii) 
(iv) 
Solution: (i) we have, 
(ii) We have, 
(iii) We have, 
(iv) We have, 
2. Write the quotient:
(i) 
(ii) 
(iii) 
(iv) 
Solution: (i)
(ii)
(iii) 
(iv) 
3. Divide and write the quotient and the remainder:
(i) 
(ii) 
(iii) 
(iv) 
Solution: (i)
Now ,
Therefore, the quotient 
and the remainder = 0
(ii)
Now ,
Therefore, the quotient 
and the remainder = – 1
(iii)
Now,
Therefore, the quotient 
and the remainder = 0
(iv)
Now,
Therefore, the quotient 
and the remainder = 0
TERMINAL EXERCISE : ALGEBRAIC EXPRESSIONS AND POLYNOMIALS
1. Mark a tick (
) against the correct alternative:
(i) The coefficient of 
in 
is
(A) 6 (B) 
(C) 
(D) 4
Answer: (C)
[The coefficient of in is 
]
(ii) Numerical coefficient of the monomial 
is
(A) 2 (B) 6 (C) 1 (D) –1
Answer: (D) – 1 .
[The numerical coefficient of the monomial of 
is – 1. ]
(iii) Which of the following algebraic expressions is a polynomial?
(A) 
(B) 
(C) 
(D) 
Answer: (A)
[ is a polynomial expression . ]
(iv) How many terms does the expression 
contain ?
(A) 5 (B) 4 (C) 3 (D) 2
Answer: (B) 4
[ There are four terms on the given expression .]
(v) Which of the following expressions is a binomial?
(A) 
(B) 
(C) 
(D) 
Answer: (D) 
[(D) is a binomial expression ]
(vi) Which of the following pairs of terms is a pair of like terms?
(A) 
(B) 
(C) 
(D) 
Answer: (C)
[ The like terms are : 
and 
]
(vii) A zero of the polynomial 
is
(A) 
(B) 
(C) 
(D) 
Answer: (B)
[ If 
, then 
]
(viii) The degree of the polynomial 
is
(A) 7 (B) 17 (C) 5 (D) 6
Answer: (A) 7
[The degree of the polynomial is 7 (= 3 +4) ]
2. Using variables and operation symbols, express each of the following verbal statements as algebraic statement:
(i) A number added to itself gives six.
(ii) Four subtracted from three times a number is eleven.
(iii) The product of two successive odd numbers is thirty-five.
(iv) One-third of a number exceeds one-fifth of the number by two.
Solution: (i) Let y be the number .
A/Q, 
(ii) Let x be the number .
A/Q, 
(iii) Let z be the number .
A/Q, 
(iv) Let x be the number .
A/Q, 
3. Determine the degree of each of the following polynomials:
(i) 
(ii) 
(iii) 
where a and b are constants. (iv) 
Where a, b and c are constants.
Solution: (i)
The degree of the polynomial of is 0 .
(ii)
The degree of the polynomial of 
is 6 (= 1+5) .
(iii) where a and b are constants.
The degree of the polynomial of 
is 3 .
(iv) Where a, b and c are constants.
The degree of the polynomial of is 4 (=2+2=3+1) .
4. Determine whether given value is a zero of the polynomial:
(i) 
(ii) 
Solution: (i)
Putting 
, then 
So, 
is a zero of the given polynomial .
(ii) 
Putting x=-1 
, then 
So, is a zero of the given polynomial .
5. Evaluate each of the following polynomials for the indicated value of the variable:
(i) 
(ii) 
Solution: (i)
Putting 
, then
(ii)
Putting 
, then
6. Find the value of 
for
and verify that the result is equal to the sum of first 10 natural numbers.
Solution: Given, 
for
Putting 
, then 
Again, 1+2+3+4+5+6+7+8+9+10 = 55 Verified.
7. Add : (i) 
(ii) 
(iii) 
(iv) 
Solution: (i) We have,
(ii) We have,
(iii) We have, 
(iv) We have,
8. Subtract : (i) 
. (ii) 
(iii) 
(iv) 
.
Solution: (i) We have, .
(ii)
(iii) We have,
(iv) We have, .
9. What should be added to 
to obtain 
?
Solution: We have, 
10. What should be subtracted from 
to obtain 
?
Solution: We have, 
11. The sum of two polynomials is 
. If one of them is 
, find the other.
Solution: We have, 
Therefore, the other polynomial is 
12. If 
, 
and 
, find B + C –A.
Solution: We have, 
13. Subtract 
from the sum of 
and 
. What is the coefficient of x in the result?
Solution: We have, , 
Now, 
Therefore, the coefficient of x is 2y .
14. Multiply : (i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
Solution: (i)
(ii) We have,
(iii) We have,
(iv) We have,
(v) We have,
(vi) We have,
(vii) We have,
(viii) 
5. Subtract the product of 
) and 
from the product of 
and 
.
Solution: We have, 
and 
Now , 
16.Divide :
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
Solution: (i) We have, 
Therefore, the quotient 
and the remainder = 0
(ii) We have, 
Now,
Therefore, the quotient = 
and the remainder = - 110
(iii) We have, 
Therefore, the quotient 
and the remainder = 0
(iv) We have, 
Therefore, the quotient 
and the remainder = 0
(v)
Now,
Therefore, the quotient 
and the remainder = – 2
(vi)
Now,
Therefore, the quotient 
and the remainder = 0 .
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