Class 10 Maths Chapter 2: Polynomials– NCERT Notes & Formula

Class 10 NCERT Maths Chapter 2: Polynomials

What Are Polynomials?

A polynomial is an algebraic expression of the form:

$$ p(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 $$

Where:

  • \( a_0, a_1, \ldots, a_n \in \mathbb{R} \)
  • \( a_n \ne 0 \)
  • \( n \in \mathbb{N} \cup \{0\} \) is the degree

Topics Covered in Chapter 2:

1. Types of Polynomials (On the basis of Degree):

  • Constant Polynomial: Degree 0 (e.g. \( p(x) = 4 \))
  • Linear Polynomial: Degree 1 (e.g. \( p(x) = 3x + 5 \))
  • Quadratic Polynomial: Degree 2 (e.g. \( p(x) = x^2 - 4x + 3 \))
  • Cubic Polynomial: Degree 3 (e.g. \( p(x) = x^3 + x^2 - 2 \))

2. Zeros of a Polynomial

Zero (or root) of a polynomial is a value of \( x \) for which \( p(x) = 0 \).

3. Relationship Between Zeros and Coefficients:

(i) For a quadratic polynomial:
Let \( p(x) = ax^2 + bx + c \), with zeros \( \alpha \) and \( \beta \).

  • Sum: \( \alpha + \beta = -\frac{b}{a} \)
  • Product: \( \alpha\beta = \frac{c}{a} \)

(ii) For a cubic polynomial:
Let \( p(x) = ax^3 + bx^2 + cx + d \), with zeros \( \alpha, \beta, \gamma \).

  • Sum: \( \alpha + \beta + \gamma = -\frac{b}{a} \)
  • Sum of products two at a time: \( \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \)
  • Product: \( \alpha\beta\gamma = -\frac{d}{a} \)

4. Division Algorithm for Polynomials:

If \( p(x) \) and \( g(x) \) are polynomials, with \( g(x) \ne 0 \), then:

$$ p(x) = g(x) \cdot q(x) + r(x) $$

Where \( q(x) \) is quotient and \( r(x) \) is remainder, and degree of \( r(x) \) is less than that of \( g(x) \).

5. Forming Polynomial from Given Zeros:

Quadratic polynomial: If zeros are \( \alpha \) and \( \beta \), then:

$$ p(x) = a(x - \alpha)(x - \beta) $$

Cubic polynomial: If zeros are \( \alpha, \beta, \gamma \), then:

$$ p(x) = a(x - \alpha)(x - \beta)(x - \gamma) $$

Important Concepts and Skills:

  • Understand and classify polynomials by degree and number of terms
  • Find zeros of a polynomial
  • Use the relationship between zeros and coefficients
  • Form polynomials from given zeros
  • Apply the division algorithm correctly

Examples:

Example 1: Zeros of a Quadratic Polynomial

Find the zeros of \( p(x) = x^2 - 5x + 6 \)

Factor:

$$ p(x) = (x - 2)(x - 3) $$

So, zeros are \( x = 2 \) and \( x = 3 \)

Example 2: Verify Relationship

For \( p(x) = x^2 - 7x + 10 \), find sum and product of zeros and verify with coefficients.

Factors:

$$ p(x) = (x - 5)(x - 2) $$

\( \alpha = 5, \beta = 2 \)

  • Sum: \( \alpha + \beta = 7 = -\frac{-7}{1} \)
  • Product: \( \alpha \beta = 10 = \frac{10}{1} \)

Example 3: Form Polynomial from Given Zeros

Find quadratic polynomial with zeros \( \alpha = 4, \beta = -3 \)

$$ p(x) = (x - 4)(x + 3) = x^2 - x - 12 $$

Example 4: Division Algorithm

Divide \( p(x) = x^3 - 3x^2 + 5x - 3 \) by \( g(x) = x - 1 \)

Use long division. You’ll get:

$$ p(x) = (x - 1)(x^2 - 2x + 3) + 0 $$

Summary Table:

Concept Description
Polynomial Algebraic expression with non-negative integer powers
Zeros Values where \( p(x) = 0 \)
Sum & Product of Zeros For quadratic: \( -\frac{b}{a}, \frac{c}{a} \)
Division Algorithm \( p(x) = g(x) \cdot q(x) + r(x) \)
Forming Polynomial Multiply \( (x - \alpha)(x - \beta) \)
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