NAVODAYA VIDYALA SAMITI

NOIDA, UP

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E- CONTENT

MATHEMATICS

CLASS X

CHAPTER 2

POLYNOMIALS

POLYNOMIALS

Algebraic Expressions : - Mathematical statements

that contain variable(s), constant(s) and

operation symbols(addition,subtraction ,multiplication

and division

Eg; 2x + y, a²+ 2ab + b², -5, 2/x, (2x + 3)

( x -1) etc.

Algebraic expressions can be

classified further as

1 ) Polynomials :- Algebraic expressions with the

exponents of the variable parts of each term as

a non-negative integer .

Eg. 2y-8, x²- 8xy + z, -15, √5 x-1 , 0 etc.

Counter examples:-

2/x, x^2/3 + 2x -5, 5 √z -3 , 2y -2

y + 3

2. Rational expressions :- Expressions of the form P(x),

Q(x)

where P(x) and Q(x) are polynomials

Eg. 2x + y, a²+ 2ab + b², -5, 2/x, (2x + 3)

x – 1

Note:- Polynomials in x are denoted as p(x), q(x), etc

and that in the variable y as p(y), q(y) etc.

TYPES OF POLYNOMIALS

Polynomials can be classified based on

a) Number of terms

b) Degree of the polynomial.

Types of polynomials based on the number of terms

1. Monomial – A polynomial with just one term. Example – 2x, 6x², 9xy

b) Binomial – A polynomial with two terms. Example – 4x²+x, 5x+4

c) Trinomial – A polynomial with three terms. Example – x²+3x+4

Degree of a polynomial :- The degree of a polynomial

is the degree of the highest degree term with non- zero

coefficient in the given polynomial.

eg. If p(x)= x³ + 2x – 10, then deg p(x) = 3,

Linear polynomial:- A linear polynomial is a polynomial

whose degree is 1.

Types of Polynomials based on Degree

.

The general form ( standard form) of a linear polynomial

in the variable x is ax + b, a≠0

A first degree polynomial is known as a linear polynomial

as the graph of it will always be a straight line.

.

For example, 2x+1 is a linear polynomial

Quadratic polynomial:- A quadratic polynomial is a

polynomial whose degree is 2

The standard form of a quadratic polynomial

in the variable x is ax²+ bx + c, a≠0.

Cubic polynomial :- A cubic polynomial is

a polynomial of degree 3.

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2x³+5x²+9x+15 is a cubic polynomial.

The general form is ax³ + bx² + cx + d, a≠0

3x²+8x+5 is a quadratic polynomial.

Value of a polynomial:- Value of a polynomial is the

value obtained when the variable takes a particular

value.

The value of the polynomial p(x) at x=a

is denoted as p(a)

Eg:- Consider the polynomial p(x) = 2x² -3x + 1.

When x= 1, we get p(1) = 2X1 – 3X1 + 1

= 2-3+1 = 0

When x=2, p(2)= 8-6 +1 = 3

When x = -3, p(-3) = 18+9 + 1= 28.

Thus every polynomial gives a unique value at a given

value of the variable.

Zero of a polynomial :-

Therefore, Zero of the polynomial 3x-6 is x= 2.

Generally the zero of the linear polynomial

ax + b is x = -b/a

Zero of a polynomial is the value of the variable

for which the value of the polynomial becomes 0.

In the example, p(x) = 3x -6 ,

p(2 ) = 3X2 -6.

ie, p(2) = 0.

PREPARED BY

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JOAN A LUKE

JNV KOLLAM

KERALA HYDERABAD REGION