NCERT Solutions for Class 10 Maths Chapter 2 Polynomials In this particular chapter you will study the geometrical interpretation of zeroes of polynomial and how to carry out divisions of the polynomials. In mathematics, there are two kinds of numbers one is the real number and another one is the imaginary numbers. NCERT solutions for class 10 maths chapter 1 real number has very detailed solutions to each and every question. This chapter will be dedicated to real numbers. Real numbers are those numbers that can be represented in the number line. It is one of the most elegant topics in NCERT class 10 maths which can be connected to real life. If anyone wants to go deeper into the number system then the real number is the first step of a mastering number system. Real numbers include the indepth classification of numbers and applications and properties related to various kinds of numbers. The number system is also very important when you will be appearing in the competitive examinations. CBSE NCERT solutions for class 10 maths chapter 1 real numbers will give you a great approach to the application of a concept in the particular question.
CBSE Class 10 maths board exam will have the following types of questions from the chapter polynomials:
Questions related to the degree of a polynomial
Questions based on the number of zeroes in a quadratic polynomial
Questions related to the sum & product of quadratic polynomial
Questions based on division algorithm
Questions on the number of zeroes in a cubic polynomial
The number of zeroes of p(x) is zero as the curve does not intersect the xaxis.
The number of zeroes of p(x) is one as the graph intersects the xaxis only once.
The number of zeroes of p(x) is three as the graph intersects the xaxis thrice.
The number of zeroes of p(x) is two as the graph intersects the xaxis twice.
The number of zeroes of p(x) is four as the graph intersects the xaxis four times.
The number of zeroes of p(x) is three as the graph intersects the xaxis thrice.
x^{2}  2x  8 = 0
x^{2}  4x + 2x  8 = 0
x(x4) +2(x4) = 0
(x+2)(x4) = 0
The zeroes of the given quadratic polynomial are 2 and 4
VERIFICATION
Sum of roots:
Verified
Product of roots:
Verified
The zeroes of the given quadratic polynomial are 1/2 and 1/2
VERIFICATION
Sum of roots:
Verified
Product of roots:
Verified
6x^{2}  3  7x = 0
6x^{2}  7x  3 = 0
6x^{2}  9x + 2x  3 = 0
3x(2x  3) + 1(2x  3) = 0
(3x + 1)(2x  3) = 0
The zeroes of the given quadratic polynomial are 1/3 and 3/2
Sum of roots:
Verified
Product of roots:
Verified
4u^{2} + 8u = 0
4u(u + 2) = 0
The zeroes of the given quadratic polynomial are 0 and 2
VERIFICATION
Sum of roots:
Verified
Product of roots:
Verified
t^{2}  15 = 0
The zeroes of the given quadratic polynomial are and
VERIFICATION
Sum of roots:
Verified
Product of roots:
Verified
3x^{2}  x  4 = 0
3x^{2} + 3x  4x  4 = 0
3x(x + 1)  4(x + 1) = 0
(3x  4)(x + 1) = 0
The zeroes of the given quadratic polynomial are 4/3 and 1
VERIFICATION
Sum of roots:
Verified
Product of roots:
Verified
The required quadratic polynomial is
The required quadratic polynomial is
The required quadratic polynomial is x^{2} + .
The required quadratic polynomial is x^{2}  x + 1
The required quadratic polynomial is 4x^{2} + x + 1
Q2 (6) Find a quadratic polynomial each with the given numbers as the sum and product of zeroes respectively. 4,1
The required quadratic polynomial is x^{2}  4x + 1
The polynomial division is carried out as follows
The quotient is x3 and the remainder is 7x9
The division is carried out as follows
The quotient is
and the remainder is 8
The polynomial is divided as follows
The quotient is and the remainder is
After dividing we got the remainder as zero. So is a factor of
To check whether the first polynomial is a factor of the second polynomial we have to get the remainder as zero after the division
After division, the remainder is zero thus is a factor of
The polynomial division is carried out as follows
The remainder is not zero, there for the first polynomial is not a factor of the second polynomial
Two of the zeroes of the given polynomial are .
Therefore two of the factors of the given polynomial are and
is a factor of the given polynomial.
To find the other factors we divide the given polynomial with
The quotient we have obtained after performing the division is
(x+1)^{2} = 0
x = 1
The other two zeroes of the given polynomial are 1.
Quotient = x2
remainder =2x+4
Carrying out the polynomial division as follows
deg p(x) will be equal to the degree of q(x) if the divisor is a constant. For example
Q5 (2) Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and deg q(x) = deg r(x)
Example for a polynomial with deg q(x) = deg r(x) is given below
example for the polynomial which satisfies the division algorithm with r(x)=0 is given below
Polynomials Excercise: 2.4
p(x) = 2x^{3} + x^{2} 5x + 2
p(1) = 2 x 1^{3} + 1^{2}  5 x 1 + 2
p(1) =2 + 1  5 + 2
p(1) = 0
p(2) = 2 x (2)^{3} + (2)^{2}  5 x (2) +2
p(2) = 16 + 4 + 10 + 2
p(2) = 0
Therefore the numbers given alongside the polynomial are its zeroes
Verification of relationship between the zeroes and the coefficients
Comparing the given polynomial with ax^{3} + bx^{2} + cx + d, we have
a = 2, b = 1, c = 5, d = 2
The roots are
Verified
Verified
Verified
p(x) = x^{3}  4x^{2} + 5x  2
p(2) = 2^{3}  4 x 2^{2} + 5 x 2  2
p(2) = 8  16 + 10  2
p(2) = 0
p(1) = 1^{3}  4 x 1^{2} + 5 x 1  2
p(1) = 1  4 + 5  2
p(1) = 0
Therefore the numbers given alongside the polynomial are its zeroes
Verification of relationship between the zeroes and the coefficients
Comparing the given polynomial with ax^{3} + bx^{2} + cx + d, we have
a = 1, b = 4, c = 5, d = 2
The roots are
Verified
Verified
Verified
Let the roots of the polynomial be
Hence the required cubic polynomial is x^{3}  2x^{2}  7x + 14 = 0
Q3 If the zeroes of the polynomial are a – b, a, a + b, find a and b.
The roots of the above polynomial are a, a  b and a + b
Sum of the roots of the given plynomial = 3
a + (a  b) + (a + b) = 3
3a = 3
a = 1
The roots are therefore 1, 1  b and 1 + b
Product of the roots of the given polynomial = 1
1 x (1  b) x (1 + b) =  1
1  b^{2} = 1
b^{2 } 2 = 0
Therefore a = 1 and .
Q4 If two zeroes of the polynomial are &nbnbsp; , find other zeroes.
Given the two zeroes are
therefore the factors are
We have to find the remaining two factors. To find the remaining two factors we have to divide the polynomial with the product of the above factors
Now carrying out the polynomial division
Now we get
So the zeroes are
The polynomial division is carried out as follows
Given the remainder =x+a
The obtained remainder after division is
now equating the coefficient of x
which gives the value of
now equating the constants
Therefore k=5 and a=5
NCERT solutions for class 10 maths  chapter wise
Chapter No. 
Chapter Name 
Chapter 1 
CBSE NCERT solutions for class 10 maths chapter 1 Real Numbers 
Chapter 2 
NCERT solutions for class 10 maths chapter 2 Polynomials 
Chapter 3 
Solutions of NCERT class 10 maths chapter 3 Pair of Linear Equations in Two Variables 
Chapter 4 
CBSE NCERT solutions for class 10 maths chapter 4 Quadratic Equations 
Chapter 5 
NCERT solutions for class 10 chapter 5 Arithmetic Progressions 
Chapter 6 

Chapter 7 
CBSE NCERT solutions for class 10 maths chapter 7 Coordinate Geometry 
Chapter 8 
NCERT solutions for class 10 maths chapter 8 Introduction to Trigonometry 
Chapter 9 
Solutions of NCERT class 10 maths chapter 9 Some Applications of Trigonometry 
Chapter 10 

Chapter 11 

Chapter 12 
Solutions of NCERT class 10 chapter maths chapter 12 Areas Related to Circles 
Chapter 13 
CBSE NCERT solutions class 10 maths chapter 13 Surface Areas and Volumes 
Chapter 14 

Chapter 15 
Once you go through all the solutions from CBSE NCERT class 10 maths chapter 2 polynomials, you will find yourself much improved in terms of concepts, concept application, and problemsolving. 90% of the questions in the board examinations come from the NCERT textbook. If you are well prepared these 90% questions then the rest 10% can be fetched by doing some extension of what you have learned in terms of concepts.
Keep working hard & happy learning!