CBSE Class 10 Maths Revision Notes Chapter 2 Polynomials 2026–27

Polynomials Class 10 explains algebraic expressions with whole number powers, their types, zeroes, graphs and coefficient relations.
For CBSE Class 10 Maths, Chapter 2 builds the base for quadratic equations, factorisation and algebraic problem-solving.

Polynomials are algebraic expressions where the powers of variables are whole numbers. In Class 10 Maths Chapter 2 Polynomials, students learn how to identify polynomials, find their degree, classify them, understand zeroes and connect zeroes with coefficients.

Use these CBSE Class 10 Maths Revision Notes Chapter 2 to revise the chapter in a simple order. Start with definitions and types, then revise zeroes, graphs, formulas, factorisation and the division algorithm for polynomials.

Key Takeaways

• Polynomial: An algebraic expression with variables having whole number powers.
• Degree: The highest power of the variable in a polynomial.
• Zero of p(x): A value of x for which p(x) = 0.
• Division Algorithm: p(x) = g(x) × q(x) + r(x), where g(x) ≠ 0.

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CBSE Class 10 Maths Chapter 2 Polynomials at a Glance

Chapter 2 Polynomials connects algebraic expressions with graphs, factorisation and formulas. These ideas help students solve questions on zeroes, coefficients and polynomial division.

Algebraic Expressions and Polynomials Class 10 Notes

Algebraic expressions use constants, variables and operations such as addition, subtraction and multiplication. A polynomial is a special algebraic expression where variables have whole number powers.

Examples of algebraic expressions:

• 3x² + 4x + 5
• 2x + 3√x
• 5xy + 7

Examples of polynomials:

• 5x³ + 3x + 1
• x² + 2x + 3
• 7x + 9

The expression 2x + 3√x is outside polynomial form because √x means x^(1/2). The power 1/2 is not a whole number.

Copyable format:

√x = x^(1/2)

2x + 3√x = 2x + 3x^(1/2)

Degree of a Polynomial in Class 10 Maths Chapter 2

The degree of a polynomial is the highest power of the variable. This helps students identify whether a polynomial is linear, quadratic or cubic.

Examples:

• Degree of x² + 2x + 3 is 2.
• Degree of x³ + 4x² + x + 8 is 3.
• Degree of 7x + 5 is 1.
• Degree of 9 is 0.

Copyable format:

Degree of x² + 2x + 3 = 2

Degree of x³ + 4x² + x + 8 = 3

Degree of 7x + 5 = 1

Degree of 9 = 0

Types of Polynomials Based on Terms and Degree

Polynomials Class 10 are classified by number of terms and by degree. This classification makes it easier to read questions and apply formulas.

Types of Polynomials Based on Number of Terms

Types of Polynomials Based on Degree

In these forms, a, b, c and d are real numbers. For quadratic and cubic polynomials, a ≠ 0.

Zeroes of a Polynomial in Polynomials Class 10

A zero of a polynomial is the value of x that makes the polynomial equal to zero. This is one of the most important ideas in Class 10 Maths Chapter 2 Polynomial Notes.

If p(x) is a polynomial and p(k) = 0, then k is a zero of p(x).

Example:

p(x) = x² - 3x + 2

For x = 1:

p(1) = 1² - 3(1) + 2

p(1) = 1 - 3 + 2

p(1) = 0

So, 1 is a zero of p(x).

For x = 2:

p(2) = 2² - 3(2) + 2

p(2) = 4 - 6 + 2

p(2) = 0

So, 2 is also a zero of p(x).

Answer:

Zeroes of x² - 3x + 2 are 1 and 2.

Graphical Meaning of Zeroes of a Polynomial

The graphical meaning of zeroes helps students connect algebra with graphs. A zero is the x-coordinate of the point where the graph of y = p(x) meets the x-axis.

For a linear polynomial, the graph is a straight line. It meets the x-axis at one point.

For a quadratic polynomial, the graph is a parabola. It can meet the x-axis at two points, touch it at one point, or stay away from it.

For a cubic polynomial, the graph can have at most three zeroes.

Copyable format:

A polynomial of degree n has at most n zeroes.

Linear polynomial = at most 1 zero

Quadratic polynomial = at most 2 zeroes

Cubic polynomial = at most 3 zeroes

Relationship Between Zeroes and Coefficients of a Quadratic Polynomial

The relationship between zeroes and coefficients helps students find the sum and product of zeroes quickly. It also helps form a quadratic polynomial when zeroes are given.

Let α and β be the zeroes of:

p(x) = ax² + bx + c, where a ≠ 0

Then:

α + β = -b/a

αβ = c/a

Copyable format:

Sum of zeroes = α + β = -b/a

Product of zeroes = αβ = c/a

In words:

Sum of zeroes = - coefficient of x / coefficient of x²

Product of zeroes = constant term / coefficient of x²

Example: Find Sum and Product of Zeroes

Find the sum and product of zeroes of:

p(x) = 9x² - 16x + 20

Compare with:

ax² + bx + c

So:

a = 9

b = -16

c = 20

Sum of zeroes:

α + β = -b/a

α + β = -(-16)/9

α + β = 16/9

Product of zeroes:

αβ = c/a

αβ = 20/9

Answer:

Sum of zeroes = 16/9

Product of zeroes = 20/9

Relationship Between Zeroes and Coefficients of a Cubic Polynomial

A cubic polynomial has degree 3. Its coefficient relations involve the sum of zeroes, product of zeroes taken two at a time and product of all zeroes.

Let α, β and γ be the zeroes of:

p(x) = ax³ + bx² + cx + d, where a ≠ 0

Then:

α + β + γ = -b/a

αβ + βγ + γα = c/a

αβγ = -d/a

Copyable format:

Sum of zeroes = α + β + γ = -b/a

Sum of product of zeroes taken two at a time = αβ + βγ + γα = c/a

Product of zeroes = αβγ = -d/a

Factorisation of Polynomials by Splitting the Middle Term

Factorisation helps students find zeroes by writing a polynomial as a product of factors. In Class 10, quadratic polynomials are often factorised by splitting the middle term.

Example:

Factorise:

2x² - 5x + 3

Split the middle term:

2x² - 5x + 3 = 2x² - 2x - 3x + 3

Group the terms:

2x² - 2x - 3x + 3 = 2x(x - 1) - 3(x - 1)

Take the common factor:

2x(x - 1) - 3(x - 1) = (x - 1)(2x - 3)

So:

2x² - 5x + 3 = (x - 1)(2x - 3)

Zeroes:

x - 1 = 0 or 2x - 3 = 0

x = 1 or x = 3/2

Answer:

Zeroes are 1 and 3/2.

Division Algorithm for Polynomials Class 10

The division algorithm for polynomials connects dividend, divisor, quotient and remainder. It is used when one polynomial is divided by another polynomial.

Let p(x) and g(x) be polynomials, where g(x) ≠ 0.

Then:

p(x) = g(x) × q(x) + r(x)

Here:

• p(x) = dividend
• g(x) = divisor
• q(x) = quotient
• r(x) = remainder

Condition:

r(x) = 0 or degree of r(x) < degree of g(x)

Copyable format:

Dividend = Divisor × Quotient + Remainder

p(x) = g(x) × q(x) + r(x)

r(x) = 0 or degree of r(x) < degree of g(x)

Steps to Divide One Polynomial by Another

• Arrange dividend and divisor in descending powers.
• Divide the highest degree term of the dividend by the highest degree term of the divisor.
• Multiply the divisor by the quotient term.
• Subtract and bring down the next terms.
• Continue until the remainder has a smaller degree than the divisor.

Polynomials Formula Table for Quick Revision

This formula table helps students revise CBSE Notes Class 10 Maths Chapter 2 before solving textbook and exam-style questions.

Solved Examples from Chapter 2 Polynomials

Solved examples show how formulas are used in actual questions. Practise them after revising the definitions and formula table.

Example 1: Find the Zeroes of a Quadratic Polynomial

Find the zeroes of:

p(x) = x² - 5x + 6

Factorise:

x² - 5x + 6 = x² - 2x - 3x + 6

x² - 2x - 3x + 6 = x(x - 2) - 3(x - 2)

x(x - 2) - 3(x - 2) = (x - 2)(x - 3)

So:

x - 2 = 0 or x - 3 = 0

x = 2 or x = 3

Answer:

Zeroes are 2 and 3.

Example 2: Verify the Relationship Between Zeroes and Coefficients

For:

p(x) = x² - 5x + 6

Zeroes are:

α = 2

β = 3

Sum of zeroes:

α + β = 2 + 3

α + β = 5

Using formula:

-b/a = -(-5)/1

-b/a = 5

So:

α + β = -b/a

Product of zeroes:

αβ = 2 × 3

αβ = 6

Using formula:

c/a = 6/1

c/a = 6

So:

αβ = c/a

The relationship is verified.

Example 3: Form a Quadratic Polynomial from Zeroes

Find a quadratic polynomial whose zeroes are 4 and 5.

Let:

α = 4

β = 5

Sum of zeroes:

α + β = 4 + 5

α + β = 9

Product of zeroes:

αβ = 4 × 5

αβ = 20

Formula:

p(x) = x² - (sum of zeroes)x + product of zeroes

So:

p(x) = x² - 9x + 20

Answer:

The required quadratic polynomial is x² - 9x + 20.

Chapter 2 Polynomials Summary for Class 10 Maths

The chapter starts with the meaning of polynomials and then moves to degree, types, zeroes, graphs and formulas. These points are useful for quick revision before tests and board practice.

• A polynomial is an algebraic expression with whole number powers of variables.
• The degree of a polynomial is the highest power of its variable.
• A linear polynomial has degree 1.
• A quadratic polynomial has degree 2.
• A cubic polynomial has degree 3.
• A zero of p(x) is a value of x for which p(x) = 0.
• Graphically, zeroes are the x-coordinates where y = p(x) meets the x-axis.
• For ax² + bx + c, the zeroes α and β follow:
  • α + β = -b/a
  • αβ = c/a
• For ax³ + bx² + cx + d, the zeroes α, β and γ follow:
  • α + β + γ = -b/a
  • αβ + βγ + γα = c/a
  • αβγ = -d/a
• The division algorithm is:
  • p(x) = g(x) × q(x) + r(x)

Useful Important Questions Class 10 Maths Links