2. Polynomials
Polynomials
2. POLYNOMIALS
SECTION = A
Q1. 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,
]
Q2. 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 
. ]
Q3. 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 
]
Q4. The product of zeroes of 
is: [SEBA 2018]
(a) 4
(b) 8
(c) 32
(d) 0
Solution : (d) 0
[ The product of zeroes 
]
Q5. A quadratic polynomial , whose zeroes are – 3 and 4 is
(a) 
(b) 
(c) 
(d) 
Solution: (c)
[ Given , 
and 
]
Q6. Which of the following expressions are polynomial ?
(a) 
(b) 
(c) 
(d) 
Solution : (c)
[ We have, 
is linear polynomial . ]
Q7. The product of the zeros of 
is [SEBA 2019]
(a) – 15
(b) 15
(c) 
(d) 
Solution : (a) – 15
[ The product of zeroes 
]
Q8. 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 
]
Q9. If one of the zeroes of the quadratic polynomial 
is – 3 , then the value of is :
(a) 
(b) 
(c) 
(d) 
Solution: (a)
[ let 
]
Q10. 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 
]
Q11. 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 . ]
Q12. 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 
]
Q13. 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 .
[ 
]
Q14. 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 .
Q15. 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
]
Q16. If 
, then the value of 
is : [ SEBA 2014]
(a) – 19
(b) – 29
(c) – 39
(d) – 49
Solution: – 39
[ We have, 
]
Q17. 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
]
Fill in the blank :
Q1. If 2 is a zero of the polynomial 
, then the value of is 
. [CBSE 2020 basic]
Solution: 1
[ let 
]
Q2. If 
is divisible by
, then the value of and
are 
and 
.
Solution: 2 and 0 .
[ We have, 
let, 
and 
]
Q3. If the sum and product of the zeroes are – 3 and 2 , then the polynomial is 
.
Solution: 
.
[ The polynomial 
]
Q4. If 
and 
are the zeroes of the quadratic polynomial 
, then the sum of zeroes is 
.
Solution: 
[ The sum of zeroes 
]
Q5. If the polynomial 
, then the value of 
is 
.
Solution : – 1 .
[ We have, 
]
Q6. If 
is one of the factors of the polynomials 
, then the remaining factor is 
.
Solution : 
.
[ We have,
]
Q7. The coefficient of 
in the polynomial 
is .
Solution: 4
[ We have, 
]
Answer following the questions :
Q1. If 3 is a zero of the quadratic polynomial 
, what is the value of 
? [SEBA 2013]
Solution: let 
Q2. The graph of 
is given below , for some polynomial . Find the number of zeroes of .
Solution : The number of zeroes is 3 , because the graph intersects the -axis at three points .
Q3. The graph of 
is given in figure , how many zeroes are there of 
?
Solution: The number of zero is 1 , because the graph intersects the -axis at one point only .
Q4. If one zero of the polynomial 
is 
, write other zero .
Solution: let other zero is .
A/Q , The sum of zeroes 
Q5. Find the quadratic polynomial whose zeros are 3 and – 4 respectively .
Solution: Here , 
and 
The quadratic polynomial 
Q6. If the divisor 
, quotient 
and remainder 
are three polynomial respectively , then find the dividend of the polynomial .
Solution: We know that , dividend 
divisor 
quotient 
remainder 
.
Section = II
Case study based questions are compulsory . Attempt any four sub parts of each question .
Each subpart carries 1 marks.
Q1. Given any polynomial and any non-zero polynomial g
, there are polynomials 
and such that 
,where 
or degree 
degree 
.
(a) If 
, then the polynomial is divisible by :
(i) 
(ii) 
(iii)
(iv) none of these .
(b) If is dividend , 
is divisor , is quotient and is remainder, then
(i) 
(ii) 
(iii) 
(iv) 
(c) If divisor , quotient and remainder are 
, 
and 
respectively , then the dividend is :
(i) 
(ii) 
(iii) 
(iv) 
(d) If 
is divisible by 
, then the value of 
is : [CBSE 2010]
(i) 7
(ii) 8
(iii) 9
(iv) 10
Solution: (a) (iii) .
(b) (iii) .
(c) (ii)
[ We have, Dividend 
]
(d) 9 [ We have , and 
]
Q2. If , and 
are the zeroes of the cubic polynomial 
, then 
, 
and 
are the factors of .
(a) If 
, 
and 
, then the cubic polynomial is :
(i) 
(ii) 
(iii) 
(iv) 
(b) If the cubic polynomial 
, then the sum of the product of its zeroes taken two at a time is :
(i) 
(ii)
(iii)
(iv)
(c) If 
and is a cubic polynomial , then the number of zeroes of is :
(i) 0
(ii) 2
(iii) 3
(iv) 4
(d) In figure, the graph of a polynomial is shown , what type polynomial represent in the graph ?
(i) linear polynomial
(ii) quadratic polynomial
(iii) cubic polynomial
(iv) constant polynomial
Solution: (a) (ii) .
[ The cubic polynomial 
]
(b) (iv)
(c) (iii) 3
(d) (iv) constant polynomial
[ because , the graph is not intersecting the axis .]
SECTION = B
Q1. 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 
.
Q2. Find the zeroes of the quadratic polynomial 
.
Solution: We have,
or 
Thus , the zeroes of the quadratic polynomial are 
and 
.
Q3. 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 .
Q4. 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 
Q5. Find a cubic polynomial whose zeroes are 2 , – 4 and – 3 respectively .
Solution: Here, 
, and 
.
The cubic polynomial 
Q6. 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 
Q7. Find the quadratic polynomial whose zeroes are 
and 
.
Solution: Here, 
and 
The quadratic polynomial
Q8. If and are the zeroes of the polynomial 
, find the value of 
. [Delhi 2013]
Solution: Since and are the zeroes of the polynomial 
respectively.
and 
Now , 
SECTION = C
Q1. Divide the polynomial by the polynomial and find the quotient and the remainder :
and 
[SEBA 2019]
Solution: Given, and
Now
Thus , the quotient 
and the remainder 
Q2. 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 .
Q3. 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 
Q4. 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.
Q5. 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 
.
Q6. State the Division algorithm for polynomials. Divide the polynomial 
by the polynomial and find the quotient and the remainder,
; 
[SEBA 2020]
Solution: If and are any two polynomials with 
, then the polynomials and such that , , where 
or degree of 
degree of .This result is known as the Division algorithm for polynomials.
Now ,
Thus, the quotient 
and the remainder 
Q7. If and are the zeroes of the polynomial 
, then form a quadratic polynomial whose zeroes are 
and
Solution: We have, 
and 
The sum of zeroes 
The product of zeroes 
The quadratic polynomial 
, where is constant .
Therefore , the quadratic polynomial is 
.
Q8. 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 .
0
We take , , 
and 
Therefore , 3 , – 1 and – are the zeroes of the cubic polynomial of .
SECTION = D
Q1. 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 
.
Q2. If a polynomial 
has zeroes as – 2 and – 3 ,then find the other zeroes .
Solution: let 
Since two zeroes are – 2 and – 3 of the polynomial .
is a factor of the given polynomial .
Now ,
Therefore, the zeroes of the given polynomial are – 2 , – 3 , – and .
Q3. Find other zeroes of the polynomial 
, if it is given that the two of the zeroes are 
and 
Solution: let 
and given the two of the zeroes areand .
is a factor of polynomial .
Now ,
, , – 3 and 2
Therefore, the zeroes of the given polynomial are 
, , – 3 and 2 .
Posted 5 years ago