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Free NCERT Class 10 Maths Chapter 2 Notes - Download PDF

Free NCERT Class 10 Maths Chapter 2 Notes - Download PDF

Updated on Jul 25, 2025 09:44 PM IST

A polynomial is an algebraic expression that contains a variable and a degree. In polynomial theory, a variable is also called an indeterminate, and the degree is known as the highest power of the variable. Polynomial expressions are made by arithmetic operations like addition, subtraction, and multiplication. These arithmetic operations are used to define the relationship between variables. Polynomials are used in many areas, like science, economics, mathematics, engineering, etc. In these areas, polynomials are considered the fundamental tool for analysis and solving problems. The main purpose of these NCERT Notes of Polynomials class 10 PDF is to provide students with an efficient study material from which they can revise the entire chapter.

This Story also Contains
  1. Polynomials Class 10 Notes: Free PDF Download
  2. NCERT Class 10 Maths Chapter 2 Notes: Polynomials
  3. Polynomials: Previous Year Question and Answer
  4. Class 10 Chapter-Wise Notes
Free NCERT Class 10 Maths Chapter 2 Notes - Download PDF
Free NCERT Class 10 Maths Chapter 2 Notes - Download PDF

After going through the textbook exercises and solutions, students need a type of study material from which they can recall concepts in a shorter time. Polynomials Class 10 Notes are very useful in this regard. In this article about NCERT Class 10 Maths Notes, everything from definitions and properties to detailed notes, formulas, diagrams, and solved examples is fully covered by our subject matter experts at Careers360 to help the students understand the important concepts and feel confident about their studies. These NCERT Class 10 Maths Chapter 2 Notes are made in accordance with the latest CBSE syllabus while keeping it simple, well-structured and understandable. For the syllabus, solutions, and chapter-wise PDFs, head over to this link: NCERT.

Polynomials Class 10 Notes: Free PDF Download

Use the link below to download the Polynomials Class 10 Notes PDF for free. After that, you can view the PDF anytime you desire without internet access. It is very useful for revision and last-minute studies.

Download PDF

NCERT Class 10 Maths Chapter 2 Notes: Polynomials

Polynomials: It is an algebraic expressions that can have an exponent as a real number or a whole number. A polynomial represented by p(x), where p is a polynomial in x, and the exponent of x is called the highest degree of x.

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Algebraic Expressions: An algebraic expression is an equation with variables and constants that is made up of arithmetic operators.

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Types of Polynomials:

Polynomials can be divided into two categories based on the following characteristics:
1. Number of terms
2. Degree of polynomials

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Based on the number of Terms:

A polynomial expression can have one or more terms, and according to the number of terms, polynomials are as follows:
1. Monomials: A polynomial that has only one term is called a monomial.
Example: 3x, 5pq, 6p2

2. Binomials: A polynomial that has two unlike terms is called a binomial.
Example: 3x + 2, 4x2 + 6

3. Trinomials: A polynomial that has three unlike terms is called a trinomial.
Example: 6x2 + 3x - 2

Based on Degree of Polynomials:

1. Zero Polynomials: A polynomial that has 0 as a degree is called a zero or constant polynomial.
Example: 2x0, 5y0

2. Linear Polynomial: A polynomial that has 1 as a degree is called a linear polynomial.
Example: 3x + 1, 4y + 5

3. Quadratic Polynomial: A polynomial that has 2 as a degree is called a quadratic polynomial.
Example: 5x2 + 3x + 1

4. Cubic Polynomial: A polynomial that has 3 as a degree is called a cubic polynomial.
Example: 7x3 + 5x - 2

Zeros of a Polynomial:

The value of the polynomials that make the polynomials equal to zero.
Example: For the polynomial p(x) = x2 - 9, the zeros are x=3 and x=−3, because p(3) = 32 - 9 = 0 and p(−3)=(−32)−9=0.

Graphical Representation

The representation of different types of polynomials is as follows.

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Geometrical Meaning of the Zeros of a Polynomial:

When the x-coordinate of the point of intersection is at the x-axis of the graph, then this is called geometrically the zeroes of a polynomial.

Geometrical Meaning of Zeros of Linear Polynomial:

The general form of the linear polynomial is y = ax + b, where a must not be equal to 0. For example, the equation is y = 3x + 1 for x = 2, and -2, the line of the graph passing through the points (2, 7) and (-2, -5)

x
2
-2
y = 3x + 1
7
-5


The graph is shown below:

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Geometrical Meaning of Zeros of a Quadratic Polynomial:

The general form of the quadratic polynomials is ax² + bx + c, where a is not equal to zero. The graph for a quadratic equation looks like a symbol U or a parabola, and the direction of U or the parabola may be upward or downward depending on the value of a. It can cut the points at zero, one, or two points.
The geometrical representation of quadratic polynomials when a > 0 and a < 0 is as follows:

According to the shape of the graph, there are three possible cases:
Case 1: When the graph cuts the x-axis at two distinct points, P and P'. Here, P and P' are the zeros of the quadratic equation ax² + bx + c.

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Case 2: When the graph cuts the x-axis at exactly one point, i.e., at two coincident
points. Therefore, the two points P and P' of Figure 1 of case 1 coincide here to become one point P. So, the x-coordinate of P is the only zero for the quadratic polynomial ax² + bx + c.

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Case 3: When the graph is either completely above the x-axis or completely below
the x-axis, then it does not cut the x-axis at any point. So, the quadratic polynomial ax2
+ bx + c has no zero in this case.

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Relationship Between Zeros and Coefficients of a Polynomial:

For quadratic Polynomials: Let α and β be the roots of the quadratic equation ax2
+ bx + c, then,
Sum of the zeroes (α + β) = −ba or -Coefficient of xCoefficient of x^2

Product of the zeroes (αβ) = ca or Constant termCoeffiecient of x^2

Example: x2 + 7x + 10
Sum of the zeroes (α + β) = −71

Product of the zeroes (αβ) = 101

For Cubic Polynomials: Let α, β and γ be the roots of a cubic polynomial ax³ + bx² + cx + d; then
α + β + γ = −ba
αβ + βγ + γα = ca
αβγ = −da

Divisor Algorithm:

If P(x), D(x), Q(x) and R(x) are the three polynomials where,
P(x) = Polynomial
D(x) = Divisor polynomial
Q(x) = Quotient polynomial
R(x) = Remainder polynomial
Therefore, the formula for the division algorithm is as follows:
P(x) = D(x) × Q(x) + R(x)

Where D(x) ≠ 0, and R(x) = 0 or the degree of R(x) < the degree of D(x).
Example: Divide the cubic polynomial 4x3 + 3x2 + 3x + 2 by the quadratic polynomial x2 + 2x + 1
The division is shown in the figure:

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Proof Using Divisor Algorithm:
P(x) = D(x) × Q(x) + R(x)
P(x) = 4x3 + 3x2 + 3x + 2
D(x) = x2 + 2x + 1
Q(x) = 4x - 5
R(x) = 9x + 7
Now, putting the values in the formula, we get:
4x3 + 3x2 + 3x + 2 = (x2 + 2x + 1) × (4x - 5) + 9x + 7
⇒ 4x3 + 3x2 + 3x + 2 = 4x3 + 8x2 + 4x - 5x2 - 10x - 5 + 9x + 7
⇒ 4x3 + 3x2 + 3x + 2 = 4x3 + 3x2 + 3x + 2
⇒ L.H.S. = R.H.S.

Polynomials: Previous Year Question and Answer

Given below are some previous year question answers of various examinations from the NCERT class 10 chapter 2, Polynomials:

Question 1: Find the zeroes of the polynomial 2x2+7x+5

Solution:
For the standard quadratic equation ax2+bx+c=0,

The sum of the roots is ba

The product of the roots is ca

Given equation, 2x2+7x+5=0.

Factors by Middle term splitting: (2x+5)(x+1)=0.

Zeroes: x=52 and x=1.

Question 2: α,β are zeroes of the polynomial 3x28x+k. Find the value of k, if α2+β2=409.

Solution:
Given α+β=83 and αβ=k3

For the standard quadratic equation ax2+bx+c=0,

The sum of the roots is ba

The product of the roots is ca

α2+β2=(α+β)22αβ=(83)22(k3)=6492k3.

α2+β2=409 (Given)

409=6492k3

2k3=649409

2k3=249

2k3=83

2k=8

k=4

Hence, the correct answer is 4.

Question 3: Find the zeroes of the polynomial p(x)=3x2+x10

Solution:

For p(x)=3x2+x10,

Standard quadratic equation: ax2+bx+c=0, for which

Sum of the roots =ba

Product of the roots = ca

We have, a=3,b=1 and c=10

Let, 3x2+x10=0

Now, let us factorise the middle term of the polynomial,

3x2+6x5x10=0

3x(x+2)5(x+2)=0

(3x5)(x+2)=0

Now, for this to be true, either (3x5)=0 or, (x+2)=0

Hence, the zeroes are: x=53 and x=2.

Class 10 Chapter-Wise Notes

All the links to chapter-wise notes for NCERT class 11 maths are given below:

NCERT Exemplar Solutions for Class 10

Students must check the NCERT Exemplar solutions for class 10 of Mathematics and Science Subjects.

NCERT Solutions for Class 10

Students must check the NCERT solutions for class 10 of Mathematics and Science Subjects.

NCERT Books and Syllabus

To learn about the NCERT books and syllabus, read the following articles and get a direct link to download them.

Frequently Asked Questions (FAQs)

1. What are polynomials in Class 10 Maths Chapter 2?

A polynomial is an algebraic expression that can have an exponent as a real number or a whole number. A polynomial represented by p(x), where p is a polynomial in x, and the exponent of x is called the highest degree of x.

2. What are the types of polynomials in NCERT Class 10 Maths?

Polynomials can be divided into two categories based on the number of terms and based on degree. Their subcategories are monomials, binomials, trinomials, zero polynomials, quadratic polynomials, and cubic polynomials.

3. What is the difference between linear, quadratic, and cubic polynomials?

Linear polynomials have a degree of 1
(example: 2x + 3)
Quadratic polynomials have a degree of 2
(Example: 2x² + 5 + 2)
Cubic polynomials have a degree of 3
(Example: 3x³ + 2x² + x + 2) 

4. What is the Remainder Theorem in polynomials?

 If P(x) is a polynomial and divides this polynomial by D(x), and get the quotient Q(x) and the polynomial remainder R(x), then,
P(x)D(x) = Q(x)+R(x)

5. How do you graph a polynomial function in Class 10?

Students can determine the value of the variable or determine the zeros of an equation, and then plot these values on the graph and create a smooth curve by connecting them.

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A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg

 

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)

K/2\,

Option 2)

\; K\;

Option 3)

zero\;

Option 4)

K/4

In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)

0.02

Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)

Molality

Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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