Polynomials Class 10 ||Maths|| Chapter 2 NCERT Notes

1. Polynomial Definition

A polynomial in one variable $x$ is an algebraic expression of the form:

$p (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}$

Where:

$a_{n}, a_{n - 1}, \dots, a_{1}, a_{0}$ are real numbers (called the coefficients of the polynomial).
$n$ is a non-negative integer, representing the degree of the polynomial.

Types of Polynomials Based on Degree:

Constant Polynomial: Degree 0 (e.g., $p (x) = 7$ )
Linear Polynomial: Degree 1 (e.g., $p (x) = 2 x + 3$ )
Quadratic Polynomial: Degree 2 (e.g., $p (x) = x^{2} - 4 x + 3$ )
Cubic Polynomial: Degree 3 (e.g., $p (x) = x^{3} + 2 x^{2} + x + 1$ )

2. Zeros of a Polynomial

The zero(s) of a polynomial $p (x)$ is any value of $x$ that satisfies $p (x) = 0$ .

For Different Degrees:

Linear Polynomial: Has exactly one zero.
Quadratic Polynomial: Can have at most two zeros.
Cubic Polynomial: Can have at most three zeros.

3. Relationship Between Zeros and Coefficients

For Quadratic Polynomials:

A general quadratic polynomial is written as:

$p (x) = a x^{2} + b x + c$

Let $α$ and $β$ be the zeros of the quadratic polynomial. The relationship between the zeros and the coefficients is:

Sum of the zeros: $α + β = - \frac{b}{a}$
Product of the zeros: $α \times β = \frac{c}{a}$

For Cubic Polynomials:

A general cubic polynomial is written as:

$p (x) = a x^{3} + b x^{2} + c x + d$

Let $α, β, γ$ be the zeros of the cubic polynomial. The relationships are:

Sum of the zeros: $α + β + γ = - \frac{b}{a}$
Sum of the product of the zeros taken two at a time: $α β + β γ + γ α = \frac{c}{a}$
Product of the zeros: $α \times β \times γ = - \frac{d}{a}$

4. Factorization of Polynomials

Polynomials can be factorized into their linear factors based on their zeros.

Quadratic Polynomials:

To factorize a quadratic polynomial $a x^{2} + b x + c$ :

Find its zeros using methods like factorization, the quadratic formula, or completing the square.
Express the quadratic as a product of linear factors: $a x^{2} + b x + c = a (x - α) (x - β)$ Where $α$ and $β$ are the zeros of the polynomial.

5. Division Algorithm for Polynomials

Given two polynomials $p (x)$ (dividend) and $g (x)$ (divisor), there exist unique polynomials $q (x)$ (quotient) and $r (x)$ (remainder) such that:

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

Where:

$r (x) = 0$ or the degree of $r (x)$ is less than the degree of $g (x)$ .

6. Algebraic Identities

Key identities related to polynomials include:

$(a + b)^{2} = a^{2} + 2 a b + b^{2}$
$(a - b)^{2} = a^{2} - 2 a b + b^{2}$
$a^{2} - b^{2} = (a - b) (a + b)$
$(x + a) (x + b) = x^{2} + (a + b) x + a b$

These identities are frequently used in factorization and simplification of polynomials.

Key Concepts:

Degree of a Polynomial: The highest power of the variable in the polynomial.
Zero of a Polynomial: The value(s) of the variable that makes the polynomial equal to zero.
Factorization: The process of expressing a polynomial as the product of its factors.
Division Algorithm: A method for dividing one polynomial by another, yielding a quotient and remainder.

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