Chapter 2 : Polynomials – Chapter-Wise Important Questions and Answers | Class 10 Mathematics CBSE Solutions
CBSE Class 10 Mathematics Chapter 2 : Polynomials Important Questions with Solutions and Answers
Chapter 2 : Polynomials
SECTION = A [ Multiple Choose Questions ]
Question: A quadratic polynomial having zeroes 7 and 0 is : [2023 C]
(a) 
(b) 
(c) 
(d) 
Solution: (d)
Here, 
The quadratic polynomial 
where 
is any real constant .
Therefore, the quadratic polynomial is 
. ]
Question: If 1 is a zero of the polynomial x ² + ax + 2, then ‘a’ is : [2024 B]
(a) 5 (b) 3 (c) – 3 (d) – 5
Solution: (c) – 3
[ Let 
]
Question: For what value of k, the product of zeroes of the polynomial 
is 2 ? [2024]
(a) 
(b) 
(c) 
(d) 
Solution: (b)
[ Let 
]
Question: If one zeroes of the polynomial 
is 1, then value of 
is : [SEBA 2017]
(a) 1 (b) – 1 (c) 2 (d) – 2
Solution: (a) 1
[ We have,
]
Question: If 
, then the value of 
is : [SEBA 2014]
(a) – 38 (b) – 39 (c) – 37 (d) – 36
Solution : (b) – 39
[Given, 
; 
]
Question: If the graph of the polynomial 
intersects 
- axis at two points , then number of zeroes of 
is : [SEBA 2016]
(a) 0 (b) 3 (c) 1 (d) 2
Solution : (d) 2
[Since, the number of zeroes is 2 , then the graph intersects the x-axis at two points .]
Question: If the cubic polynomial 
, then the sum of its zeroes taken two at a time is :
(a) 
(b) 
(c) 
(d) 
Solution: (d)
[ 
The sum of its zeroes taken two at a time 
. ]
Question: If one zero of the quadratic polynomial 
is 2 , then the value of 
is
(a) 10 (b) – 10 (c) 5 (d) – 5
Solution: (b) – 10
[ let 
]
Question: The product of zeroes of 
is: [SEBA 2018]
(a) 4 (b) 8 (c) 32 (d) 0
Solution : (d) 0
[ The product of zeroes 
]
Question: The graph of a polynomial intersects the y-axis at one point and the x-axis at two points. The number of zeroes of this polynomial are : [2024 S]
(a) 1 (b) 2 (c) 3 (d) 0
Answer: (b) 2
[ The number of zeroes of a polynomial is equal to the number of points where its graph intersects the x-axis, because zeroes are the x-values where y = 0 .
So, the polynomial has two zeroes. ]
Question: A quadratic polynomial , whose zeroes are – 3 and 4 is
(a) 
(b) 
(c) 
(d) 
Solution: (c)
[ Given , 
and 
]
Question: Which of the following expressions are polynomial ?
(a) 
(b) 
(c) 
(d) 
Solution : (c)
[ We have, 
is linear polynomial . ]
Question: The product of the zeros of 
is [SEBA 2019]
(a) – 15 (b) 15 (c) 
(d) 
Solution : (a) – 15
[ The product of zeroes 
]
Question: If the zeroes of the quadratic polynomial 
are 2 and – 3 , then
(a) 
(b) 
(c) 
(d) 
Solution: (d) 
[ let 
A/Q , Sum of zeroes 
and product of zeroes 
]
Question: If one of the zeroes of the quadratic polynomial 
is – 3 , then the value of is :
(a) 
(b) 
(c) 
(d) 
Solution: (a)
[ let 
]
Question: The sum of the zeroes of the cubic polynomial 
is : [SEBA 2015]
(a) 5 (b) 11 (c) 3 (d) 
Solution: (d)
[ The sum of the zeroes 
]
Question: If the graph of the polynomial 
intersects 
- axis at two points , then number of zeroes of 
is : [SEBA 2016]
(a) 0 (b) 3 (c) 1 (d) 2
Solution : (d) 2
[ The number of zeroes is 2 as the graph intersects the 
axis at two points . ]
Question: If one of the zeroes of the cubic polynomial 
is – 1 , then the product of the other two zeroes is :
(a) 
(b) 
(c) 
(d) 
Solution: (d) 
[ 
The product of zeroes 
]
Question: The graph of the polynomial 
is a downward parabola , if [2025 B]
(a) a > 0 (b) a < 0 (c) a = 0 (d) a > 1
Answer: (b) a < 0
[ , when a < 0 , the parabola opens downward because the coefficient of x² is negative.]
Question: The zeroes of the quadratic polynomial 
are :
(a) both positive (b) both negative (c) One positive and one negative (d) both equal
Solution: (c) One positive and one negative .
[ 
]
Question: If 
is a polynomial of at least degree one and 
, then 
is know as :
(a) The value of 
. (b) Zero of 
. (c) Constant term of 
(d) none of these
Solution: (b) Zero of .
Question: Consider the following statements :
(i) 
is a factor of 
(ii) 
is a factor of 
(iii) 
is a factor of 
In these statements :
(a) (i) and (ii) are correct (b) (i) , (ii) and (iii) are correct (c) (ii) and (iii) are not correct (d) (i) and (iii) are correct
Solution: (c) (ii) and (iii) are not correct
[ (i) is a factor of
(ii) is a factor of
(iii) is a factor of
]
Question: If 
, then the value of 
is : [ SEBA 2014]
(a) – 19 (b) – 29 (c) – 39 (d) – 49
Solution: – 39
[ We have, 
]
Question: On dividing a polynomial by
, quotient and remainder are found to be
and 
respectively . The polynomial is : [ CBSE 2020 standard]
(a) 
(b) 
(c) 
(d) 
Solution : (b)
[ 
divisor 
quotient 
remainder
]
Question: If 
are zeroes of the polynomial 
,then the value of 
is [2025 B]
(a) – 4 (b) 
(c) 
(d) 4
Solution: (b)
[ 
Now, 
]
Question: The graph of 
is given below , for some polynomial . Find the number of zeroes of .
(a) 2 (b) 3 (c) 4 (d) 1
Solution : (b) 3
[ The number of zeroes is 3 , because the graph intersects the -axis at three points . ]
Question: If one of the zeroes of the quadratic polynomial 
is 2 , then the value of k is [2020 ]
(a) 10 (b) – 10 (c) – 7 (d) – 2
Answer: (b) – 10
[Let, 
Question: The graph of 
is given in figure , how many zeroes are there of 
?
(a) 0 (b) 3 (c) 2 (d) 1
Solution: (d) 1
[ The number of zero is 1 , because the graph intersects the -axis at one point only . ]
Fill in the blank :
Question: If 2 is a zero of the polynomial 
, then the value of is 
. [CBSE 2020 basic]
Solution: 1
[ let 
]
Question: If 
is divisible by
, then the value of and
are 
and 
.
Solution: 2 and 0 .
[ We have, 
let, 
and 
]
Question: If the sum and product of the zeroes are – 3 and 2 , then the polynomial is 
.
Solution: 
.
[ The polynomial 
]
Question: If 
and 
are the zeroes of the quadratic polynomial 
, then the sum of zeroes is 
.
Solution: 
[ The sum of zeroes 
]
Question: If the polynomial 
, then the value of 
is 
.
Solution : – 1 .
[ We have, 
]
Question: If 
is one of the factors of the polynomials 
, then the remaining factor is 
.
Solution : 
.
[ We have,
]
Question: The coefficient of 
in the polynomial 
is .
Solution: 4
[ We have, 
]
Answer Following the Questions :
Question: For what value of k , 3 is a zero of the polynomial 
? [2010]
Solution : Let 
Question: If 3 is a zero of the quadratic polynomial 
, what is the value of 
? [SEBA 2013]
Solution: let 
Question: If 
are the zeroes of a polynomial 
, write the value of 
. [2010]
Solution : We have,
or 
Now, 
Question: If one zero of the polynomial 
is 
, write other zero .
Solution: let other zero is .
A/Q , The sum of zeroes 
Question: If sum of the zeroes of the quadratic polynomial 
is 3 , then find the value of k .[2009]
Solution : We have,
We know that, 
Therefore, the value of k is 9 .
Question: Find the quadratic polynomial whose zeros are 3 and – 4 respectively .
Solution: Here , 
and 
The quadratic polynomial 
SECTION = B
Question: Find the quadratic polynomial whose sum and product of the zeroes are 
and 
. [2012]
Solution : Given, 
and 
We know that , the quadratic polynomial 
, where, k is a constant .
Therefore, the quadratic polynomial is 
.
Question: For what value of 
, 
is the zero of the polynomial 
? Also, the other zeroes of the polynomial.
Solution: let 
So, 
or 
Thus, the zeroes of the given polynomial are – 7 and 
.
Question: Find the zeroes of the quadratic polynomial 
.
Solution: We have,
or 
Thus , the zeroes of the quadratic polynomial are 
and 
.
Question: Find a quadratic polynomial the sum and product of whose zeroes are – 7 and 10 respectively. Hence find the zeroes .
Solution: The quadratic polynomial 
We have , 
or 
Therefore, the zeroes of are – 2 and – 5 .
Question: Find a cubic polynomial with the sum , sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2 , – 7 and – 14 respectively.
Solution: We know that , the cubic polynomial
Therefore , the polynomial is 
Question: Find a cubic polynomial whose zeroes are 2 , – 4 and – 3 respectively .
Solution: Here, 
, and 
.
The cubic polynomial 
Question: If two zeroes of the polynomial 
are 
and 
,then find its third zero. [CBSE 2010]
Solution: let 
be the third zero .
Here , 
and 
.
The sum of the zeroes 
Question: Find the quadratic polynomial whose zeroes are 
and 
.
Solution: We have,
and 
The quadratic polynomial 
, where k is a constant .
Therefore, the quadratic polynomial is 
.
Question: If and are the zeroes of the polynomial 
, find the value of 
. [Delhi 2013]
Solution: Since and are the zeroes of the polynomial 
respectively.
Now , 
Question: Find the zeroes of the quadratic polynomial 
. [2013]
Solution : We have, 
or 
Therefore, the zeroes of the quadratic polynomial are 
and 
.
SECTION = C
Question: If 
and
are the zeroes of the polynomial 
, find the value of 
[2012 ]
Solution : We have,
and 
Now, 
Question: Find the Zeroes of the polynomial 
and verify the relationship between the zeroes and the coefficients . [CBSE 2014]
Solution : Let , 
or 
Therefore, the zeroes of the polynomial 
are 
and .
The sum of its zeroes 
The product of its zeroes 
Verified.
Question: Find the zeroes of the polynomial 
and verify the relationship between the zeroes and the coefficients .
Solution: We have,
or 
Therefore , the zeroes of the given polynomial 
and 
.
The sum of the zeroes 
The product of the zeroes 
verified .
Question: If 
and
are the zeroes of the quadratic polynomial 
such that
, find the values of k . [2013]
Solution : Given, 
and 
A/Q, 
or 
Therefore, the value of k are – 1 and 
Question: Verify that 4 , – 2 , 
are the zeroes of the cubic polynomial 
and then verify the relationship between the zeroes and the coefficients .
Solution: let 
Here , 
, 
and 
So, 
and 
Verified.
Question: Find the quadratic polynomial whose zeroes are – 2 and – 5 . Verify the relationship between zeroes and coefficients of the polynomial. [ Delhi 2013 , SEBA 2018]
Solution: Here, 
and 
The quadratic polynomial 
, where 
is a constant .
Therefore, the quadratic polynomial is .
Now , the sum of zeroes 
The product of zeroes 
verified.
Question: If the zeroes of the polynomial 
are
, 
, 
, find and .
Solution: Since , 
, and 
are the zeroes of the polynomial .
A/Q , the sum of the zeroes 
and product of zeroes 
Therefore , the value of 
and 
are 1 and 
.
Question: If the zeroes of the polynomial 
are double in the value to the zeroes of 
, find the value of 
and
. [CBSE 2012]
Solution : Let 
and 
are the zeroes of the polynomial respectively.
We have, 
and 
For the polynomial 
.
and 
From 
and 
, we get 
From 
and 
, we get 
Question: If and are the zeroes of the polynomial 
, then form a quadratic polynomial whose zeroes are 
and 
.
Solution: We have, 
The sum of zeroes 
The product of zeroes 
The quadratic polynomial 
, where 
is constant .
Therefore , the quadratic polynomial is 
.
Question: If the sum of squares of the zeroes of the polynomials 
is 
. Find the value of
. [CBSE 2015]
Solution : Let , and are the zeroes of the polynomial 
.
and 
Given, 
Question: Verify that 3 , – 1 , 
are the zeroes of the cubic polynomial 
and then verify the relationship between the zeroes and the coefficients .
Solution: let and given, 3 , – 1 , are the zeroes of .
We take , , 
and 
Therefore , 3 , – 1 and are the zeroes of the cubic polynomial of .
SECTION = D
Question: If the zeroes of the polynomial 
are , and 
, find and as well as the zeroes of the given polynomial.
Solution: The polynomial is
The sum of its zeroes 
The product of its zeroes 
[ From 
]
Putting the value of 
and 
in equation , we get
and 
Therefore, the value of and are 
or 
and 
or 
.
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