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Free NCERT Class 10 Maths Chapter 2 Notes - Download PDF

Free NCERT Class 10 Maths Chapter 2 Notes - Download PDF

Edited By Ramraj Saini | Updated on Apr 01, 2025 02:27 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 as the fundamental tool for analysis and solving problems.
These notes contain all the topics, subtopics, short tricks, formulae, and all the required important key points. These notes cover the basic definition of polynomials, algebraic expressions, degree of polynomials, and types of polynomials and their graphical representation. CBSE Class 10 chapter Polynomials also includes the geometrical representation of the zeroes of polynomials. Students must practice all the topics of polynomials and their examples from the NCERT Exemplar Solutions for Class 10 Maths Chapter 2 Polynomials. Students must practice questions and check their solutions using the NCERT Solutions for Class 10 Maths Chapter 2 Polynomials.

This Story also Contains
  1. NCERT Class 10 Maths Chapter 2 Polynomials: Notes
  2. Types of Polynomials:
  3. Graphical Representation
  4. Relationship Between Zeros and Coefficients of a Polynomial:
  5. Divisor Algorithm:
  6. Class 10 Chapter-Wise Notes
  7. NCERT Class 10 Exemplar Solutions for Maths And Science

NCERT Class 10 Maths Chapter 2 Polynomials: Notes

Polynomials: It 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.
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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.

Class 10 Chapter-Wise Notes

Students must download the notes below for each chapter to ace the topics.

NCERT Class 10 Solutions for Maths And Science

Students must check the NCERT solutions for Class 10 Maths and Science given below:

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NCERT Class 10 Exemplar Solutions for Maths And Science

Students must check the NCERT exemplar solutions for Class 10 Maths and Science given below:

Read More: Syllabus and Books

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. How do you find the zeros of a polynomial?

Students can determine zeros of polynomials by setting the  equation equal to zero and determining the factors of the equation, and solve them or by applying the formula of sum of the zeroes (α + β) = ba or -Coefficient of xCoefficinet of x^2 and product of the zeroes (αβ) = ca or Constant termCoeffiecinet of x^2 

5. What is the relationship between zeros and coefficients of a quadratic polynomial?

For quadratic equation the sum of the zeroes (α + β) = ba or -Coefficient of xCoefficient of x^2 and product of the zeroes (αβ) = ca or Constant termCoeffiecient of x^2

Articles

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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