NCERT solutions for class 10 maths chapter 2 Polynomials  In the previous class, you have already studied the polynomial with one degree. In class 10 maths chapter 2 you will study the polynomials with any degree and its solutions using graphical representation. Solutions of NCERT for class 10 maths chapter 2 Polynomials have detailed explanations to each and every question. A polynomial is constructed using variables and numbers using addition, subtraction, multiplication, and division. When two polynomials are divided then the resultant is called the rational expression of polynomials. NCERT solutions for class 10 maths chapter 2 Polynomials is an important tool to score high in the board examinations. Polynomial is an important part of algebra in which we get the knowledge about the various operation performed over polynomials. Polynomials are further classified according to their degrees. A degree in the polynomials is the highest power of variable present in the equation. Let's understand the degree by taking some examples
In the equations, the maximum power of the variables is 1 and 2 respectively. So the degree for the polynomials is 1 and 2 respectively. Polynomials with degrees 1, 2, 3 are known as linear, quadratic and cubic polynomials respectively. CBSE NCERT solutions for class 10 maths chapter 2 Polynomials is covering each question in a detailed manner. Apart from this particular chapter, you can also download the solutions classwise and subject wise by clicking on the link where NCERT solutions are there.
The degree of a polynomial
Number of zeroes in a quadratic polynomial
The sum & product of quadratic polynomial
Division algorithm
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
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
NCERT solutions for class 10 maths chapter 2 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 , 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
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 
You will find yourself much improved in terms of concepts, their applications, and problemsolving. 90% of the questions in the board examinations come from the NCERT syllabus.
Now when you have gone through the solutions of NCERT class 10 maths chapter 2 polynomials, you must have learnt to answer a question in the step by step manner.
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Q1 (ii) Find the zeroes of the following quadratic polynomials and verify the relationship between
the zeroes and the coefficients.
Q1 (4) Find the zeroes of the following quadratic polynomials and verify the relationship between
the zeroes and the coefficients.
Q1 (1) Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder
in each of the following :