The document Short Notes - Polynomials Class 9 Notes | EduRev is a part of the Class 9 Course Mathematics (Maths) Class 9.

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**Facts that Matter **

- An algebraic expression, in which the variables involved have only whole number powers, is called a polynomial.
- Polynomials of degree 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.
- Polynomials containing one, two and three non-zero terms are called monomial, binomial and trinomial respectively.
- If f(x) be a polynomial of degree n ≥ 1 and let ‘a’ be any real number, then f(a) is the remainder for ‘f(x) being divided by (x – a)’.
- Let f(x) be a polynomial of degree n ≥ 1, then (x – a) is a factor of f(x) provided f(a) = 0. Also if (x + a) is a factor of f(x), then f(–a) = 0
- x
^{3}+ y^{3}= (x + y)(x^{2}– xy + y^{2}) - x
^{3}– y^{3}= (x – y)(x^{2 }+ xy + y^{2}) - x
^{3}+ y^{3}+ z^{3}– 3xyz = (x + y + z)(x^{2}+ y^{2}+ z^{2}– xy – yz – zx) - If x + y + z = 0, then x
^{3}+ y^{3}+ z^{3 }= 3xyz - (x + y + z)
^{2}= x^{2}+ y^{2 }+ z^{2}+ 2xy + 2yz + 2zx - (x + y)
^{3}= x^{3}+ y^{3 }+ 3xy (x + y) - (x – y)
^{3}= x^{3}– y^{3}– 3xy (x – y)

**VARIABLE**

A symbol which can be assigned different numerical values is known as a variable. Variables are generally denoted by x, y, z, p, q, r, s, etc.

**CONSTANT**

A symbol having a fixed value is called a constant, e.g. 8, 5, 9, p, a, b, c, etc. are constants.**Note: **The values of constants remain the same throughout a particular situation, but the value of a variable can keep changing.

**ALGEBRAIC EXPRESSION**

A combination of constants and variables, connected by some or all the basic operations +, –, x, is called an algebraic expression.

**Example**: is an algebraic expression.**TERMS**

Various parts of an algebraic expression separated by (+) or (–) operations are called terms.

Examples: (i) In the above algebraic expression, terms are: 7, 8x, –6x^{2}y and

(ii) Various terms of 3p^{4} – 6q^{2} + 8r^{3}s – 2pq + 6s^{3} are: 3p^{4}, –6q^{2}, 8r^{3}s, –2pq and 6s^{3}

**POLYNOMIAL**

An algebraic expression in which the variables involved have non-negative integral powers is called a polynomial.

**Examples: **(i) x^{3} + x^{2} – 4x – 7;

(ii) 3p^{3} + 5p – 9;

(iii) x^{2} + 2x; etc. are all polynomials

**Note: I. **Each variable in a polynomial has a whole number as its exponent.

II. Each term of a polynomial has a co-efficient.

**DEGREE OF A POLYNOMIAL**

I . In case of a polynomial involving in one variable, the highest power of the variable is called the degree of the polynomial.**Example: **The degree of x^{5} – 2x^{3} + x is 5.

II. In case of a polynomial involving in more than one variable, the highest sum of exponents of variables in any term is called the degree of the polynomial.**Example: **The degree of p^{2} – 6p^{6}q + 5p^{2}q^{3} – 3q^{4} is 7

(∵ The term –p^{6}q has the sum of exponents of p and q as 6 + 1, i.e. 7)**TYPES OF POLYNOMIALS**

A [On the basis of number of its terms]

**(i) Monomial:** A polynomial containing only one non-zero term is called a monomial.

**(ii) Binomial:** A polynomial containing two non-zero terms is called a binomial.

**(iii) Trinomial:** A polynomial containing three non-zero terms is called a trinomial.

Note: I. A monomial containing a constant only is called a constant polynomial.

II. A monomial containing its term as zero only is called a zero polynomial.

III. If we add polynomials, we get a polynomial.

IV. If we multiply polynomials, we get a polynomial.**B [On the basis of its degree] **

**(i) Linear Polynomial:** A polynomial of degree 1 is called a linear polynomial.

**(ii) Quadratic Polynomial: **A polynomial of degree 2 is called a quadratic polynomial.

**(iii) Cubic Polynomial:** A polynomial of degree 3 is called a cubic polynomial.

**(iv ) Biquadratic Polynomial:** A polynomial of degree 4 is called a biquadratic polynomial.

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