CBSE Notes Class 10 Maths Chapter 2: Polynomials

The CBSE Notes for Class 10 Maths for Chapter 2 Polynomials help students learn about polynomials and become proficient in algebraic expressions. The most recent CBSE curriculum, which serves as the basis for these lectures, breaks down each subject into easily understood principles, formulas, and examples for practice on tests. Students are prepared for more complex algebraic topics in the higher courses by gaining a fundamental grasp of polynomials, types, and operations in Chapter 2.

1.0CBSE Notes for Class 10 Maths Chapter 2: Polynomials - Free PDF!!

Students can download CBSE Notes for Class 10 free PDF notes for Maths second chapter, Polynomials. This readily accessible resource is designed to facilitate easy understanding of the concepts within the Polynomials chapter.

2.0CBSE Class 10 Maths Chapter 2 Polynomials - Revision Notes

This chapter introduces the idea of polynomials. Expressions that contain exponents, constants, and variables are called polynomials. In algebra, it is a crucial issue. The following lists some of the key ideas that lie under the umbrella of "polynomials":

Important Concepts in Polynomials

Definition of Polynomials: Expressions that include variables with non-negative integer exponents, such as $3x^2+2x-5.$

1. Types of Polynomials

• Constant Polynomial: Contains no variables, only a constant (e.g., 5, 7, 10).
• Linear Polynomial: Has a variable raised to the power of 1 (e.g., x + 2 or 3x + 5).
• Quadratic Polynomial: Contains a variable raised to the power of 2 (e.g. $x^2+2x+1$).
• Cubic Polynomial: Contains a variable raised to the power of 3 ($e.g.,x^3-3x^2+2$).
• Zeros of a Polynomial: The values of the variable that make the polynomial equal to zero. For example, in x² − 4, the zeros are x = 2 and x = −2.

2. Definitions:

• Degree of a Polynomial: The highest power of the variable in a polynomial. For example, x³ + 3x + 1 has a degree of 3.
• Monomial, Binomial, and Trinomial: These refer to polynomials with one, two, and three terms, respectively, like 3x (monomial), x + 1 (binomial), and x² + 2x + 1 (trinomial).

3. Formulas:

• Standard Form of a Quadratic Polynomial: $a{x}^2+bx+c=0$, where a, b, and c are constants.
• Finding Zeros of a Polynomial: For a quadratic polynomial $a{x}^2+bx+c=0$, the zeros can be found using the formula.

$\mathrm{X}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

• Relationship between Zeros and Coefficients:

 For ax² +  bx + c: Sum of zeros = $-\frac{b}{a}$

Product of zeros = $\frac{c}{a}$

4. Tips & Tricks:

• Remember Polynomial Degrees: Constant (0), Linear (1), Quadratic (2), and Cubic (3).
• Using the Factor Theorem: This can help to quickly verify if a given value is a zero of a polynomial by substituting it into the polynomial equation.
• Using Graphs for Visual Understanding: Plotting polynomials can help in understanding the nature of roots (real or imaginary) and the shape of the curve.

3.0CBSE Class 10 Maths Chapter 2 Polynomials - Key Notes

Polynomials

• An algebraic expressions of form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + aₙ₋₂xⁿ⁻² + …….. + a₁x + a₀.
• a₀, a₁, ….. aₙ are real numbers and all indices of variable x are non-negative integers.
• a₀, a₁x, ……..aₙxⁿ are terms and a₀, a₁, ……..aₙ are coefficients.
• For e.g. p(x) = 6x³ + 4x² + 3x + 1

Degree of Polynomial

• Highest power of variable is called degree.
• For e.g. The degree of the polynomial 6x⁴ + 2x³ + 3 is 4.

Types of Polynomials

ON BASIS OF TERMS

Monomial (2x)
Binomial (x² + 3x)
Trinomial (3x³ − x² + 3x)
Quadrinomial (2x⁴ + 3x³ − x² + 3x)

ON BASIS OF DEGREE

Zero (0)
Constant (3)
Linear (2x + 1)
Quadratic (x² − 3)
Cubic (x³ − 4x + 3)
Bi-Quadratic (x⁴ − x + 5)

Zero or Root of polynomial

• x = r is a zero of a polynomial p(x), if p(r) = 0
• For e.g. Check whether 2 and 3 are zero of given polynomial
  p(x) = x² − 5x + 6 or not.

Sol.
p(2) = (2)² − 5 × 2 + 6 = 4 − 10 + 6 = 0
p(3) = (3)² − 5 × 3 + 6 = 9 − 15 + 6 = 0

∴ 2 and 3 are the zeros of the polynomial p(x).

Value of Polynomial

The value of a polynomial f(x) at x = a is obtained by substituting x = a in given polynomial.

For e.g. Find the value of polynomial 5x − 4x² + 3 at x = −1.

Sol. p(x) = 5x − 4x² + 3

To find value of polynomial at x = −1 or p(−1)

So we put x = −1 in polynomial,

p(−1) = 5 × (−1) − 4 × (−1)² + 3
p(−1) = −5 − 4 + 3
p(−1) = −6

So value of polynomial at x = −1 is −6.

Remainder Theorem

If p(x) is divided by (x − a) then remainder is given by p(a).

For e.g. Find the remainder when polynomial p(x) = x³ − 4x² − 7x + 10 is divided by x − 2.

Sol. 

p(x) = x³ − 4x² − 7x + 10

p(2) = 2³ − 4(2)² − 7(2) + 10

= 8 − 16 − 14 + 10 = −12

Factor Theorem

• If (x − a) is a factor of p(x), then p(a) = 0 and vice versa.
• For e.g. Examine whether x + 2 is a factor of x³ + 3x² + 5x + 6.

Sol. We know that the zero of the polynomial (x + 2) is −2.

Let p(x) = x³ + 3x² + 5x + 6

Then,
p(−2) = (−2)³ + 3(−2)² + 5(−2) + 6
= −8 + 12 − 10 + 6 = 0

Hence x + 2 is factor of given polynomial.

• According to the factor theorem, if p(a) = 0, then (x − a) is a factor of p(x).

Geometrical meaning of the zeros of a polynomial

• Geometrical representation of the zero of a linear polynomial y = ax + b

• Geometrical representation of the zero of a quadratic polynomial y = ax² + bx + c

Case-I : Two distinct zeros

Case-II : One zero

Case-III : No zero

Relationship between the zeros and coefficients of a polynomial

For a linear polynomial

Zero of a linear polynomial = − b / a= - (constant term / coefficient of x)

For a quadratic polynomial

• Sum of zeros = α + β = − b / a= - (coefficient of x / coefficient of x²)
• Product of zeros = αβ = c / a = - (constant term / coefficient of x²)

Then polynomial f(x) is given by f(x) = K{x² − (α + β)x + αβ}

For a cubic polynomial

Sum of zeros = α + β + γ = − b / a = - (coefficient of x² / coefficient of x³)

Sum of product of 2 zeroes at a time = αβ + βγ + γα = c / a = (coefficient of x / coefficient of x³)

Product of zeros = αβγ = − d / a =  (constant term / coefficient of x³)

Polynomial f(x) is given by f(x) = K [x³ − (α + β + γ)x² + (αβ + βγ + γα)x − αβγ]

Division algorithm for polynomials

If f(x) is a polynomial and g(x) is a non-zero polynomial, then there exist two polynomials q(x) and r(x) such that f(x) = g(x) × q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x). In other words,

Dividend = Divisor × Quotient + Remainder

Remark: If r(x) = 0, then polynomial g(x) is a factor of polynomial f(x).

4.0Symmetric functions of the zeros

Some useful relations involving α and β are

α² + b² = (α + b)² − 2αb

(α − b)² = (α + b)² − 4αb

α² − b² = (α + b)(α − b) = (α + b) √((α + b)² − 4αb)

α³ + b³ = (α + b)³ − 3αb (α + b)

α³ − b³ = (α − b)³ + 3αb (α − b)

α⁴ − b⁴ = (α² + b²) (α + b) (α − b)= [(α + b)² − 2αb] (α + b) √((α + b)² − 4αb)

α⁴ + b⁴ = (α² + b²)² − 2(αb)²= [(α + b)² − 2αb]² − 2(αb)²

α⁵ + b⁵ = (α³ + b³) (α² + b²) − α²b² (α + b) = [(α + b)³ − 3αb (α + b)] [(α + b)² − 2αb] − (αb)² (α + b)

5.0Algebraic Identities

(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
a² − b² = (a + b)(a − b)
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
(a + b)³ = a³ + b³ + 3ab(a + b)
(a − b)³ = a³ − b³ − 3ab(a − b)
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)
a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

Special case: If a + b + c = 0 then a³ + b³ + c³ = 3abc

6.0Key Features of CBSE Maths Notes for Class 10 Chapter 2

• Exam-Focused Content: These are useful revision notes since they include important exam-related formulae, properties, and theorems that assist pupils in concentrating on scoring portions.
• Solved Examples for Better Understanding: These revision notes are helpful since they include solved examples along with detailed answers, which will assist the learner in comprehending how polynomial issues may be solved.
• Formula Tables for Quick Reference: Being in easy-to-read table format, these key formulas would be useful in finding the sum and product of zeroes, polynomial identities, and algorithms on division are very helpful towards making last-minute revision much more efficient.
• Simple Language and Illustrations: Simple, everyday language is used to explain concepts, and images are used to aid in explaining more complicated ideas, such as polynomial graphing.
• Visual Aids and Graphical Representations: To help comprehend the nature of roots and how they affect the structure of the polynomial graph, graphs and other visual aids are used to show how polynomial functions behave. This is especially beneficial for those who learn best visually.
• Aligned with CBSE Syllabus: The notes are created to meet CBSE standards, ensuring that all important topics in Chapter 2 are covered comprehensively.

For students looking to improve their algebraic skills, these CBSE Maths Notes for Class 10 Chapter 2 on Polynomials are a great resource. By emphasising terminology, formulae, and solved problems, the notes help students become more comfortable answering questions pertaining to polynomials on tests.