# Polynomials Class 9 Notes Maths - Chapter 2

## Chapter: 2

## What Are Polynomial ?

## Meaning Of A Polynomial

^{2}– y

^{2}+ 2xy, x

^{2}+ 2, etc.

- A polynomial in one variable, say x, is an algebraic expression of the form P(x) = a
_{n}x^{n}+ a_{n-1}x^{n-1}+.......a_{2}x^{2}+ a_{1}x + a_{0}, where a_{0}, a_{1}, a_{2},.......,a^{n are constants and respectively known as coefficients of x0, x, x2......., xn.}

## Terms Of A Polynomial

_{n}x

^{n}+ a

_{n-1}x

^{n-1}+ ..... + a

_{2}x

^{2}+ a

_{1}x + a

_{0}, then each of a

_{n}x

^{n}, a

^{n-1}x

^{n-1},......, a

_{0}is called a term of the polynomial p(x).

## Degree Of A Polynomial

## Types Of Polynomials

**According to Number of Terms:**

• Polynomials having only one term are called monomials.

• Polynomials having only two terms are called binomials.

• Polynomials having only three terms are called trinomials.

**According to Degree:**

• A polynomial of degree 0 is called constant polynomial.

• A polynomial of degree 1 is called linear polynomial.

• A polynomial of degree 2 is called quadratic polynomial.

• A polynomial of degree 3 is called cubic polynomial.

• A polynomial of degree 4 is called biquadratic polynomial.

## Zeroes Of A Polynomial

• Any real number `k’ is called a zero of a polynomial p(x), if p(K) = 0

• If `k’ is the zero of p(x), then we can also say `k’ is a root of the equation p(x) = 0

**Some Important Facts**

• Every linear polynomial in one variable has a unique zero.

• Every non-zero constant polynomial has no zero.

• Every real number is a zero of the zero polynomial

## Remainder Theorem

If p(x) is any polynomial of degree n (n ≥ 1) and p(x) is divided by the linear polynomial x – a, then the remainder is p(a).

## Factor Theorem

x – a is a factor of the polynomial p(x), if p(a) = 0 and vice versa, i.e. if x – a is a factor of p(x), then p(a) = 0

## Algebraic Identities

^{2}= x

^{2}+ y

^{2}+ 2xy

^{2}= x

^{2}+ y

^{2}– 2xy

^{2}– y

^{2}= (x + y) (x – y)

• (x + a) (x + b) = x2 + (a + b) x + ab

^{2}= x

^{2}+ y

^{2}+ z

^{2}+ 2xy + 2yz + 2zx

^{3}= x

^{3}+ y

^{3}+ 3xy (x + y) = x

^{3}+ y

^{3}+ 3x

^{2}y + 3xy

^{2}

^{3}= x

^{3}– y

^{3}– 3xy (x – y) = x

^{3}– y

^{3}– 3x

^{2}y + 3xy

^{2}

^{3}+ y

^{3}= (x + y) (x

^{2}+ y

^{2}– xy)

^{3}– y

^{3}= (x – y) (x

^{2}+ y

^{2}+ xy)

^{3}+ y

^{3}+ z

^{3}– 3xyz = (x + y + z) (x

^{2}+ y

^{2}+ z

^{2}– xy – yz – zx)