Presented To : Mrs.Shewta MamPresented By: Pragti Jain

Class:Xth

POLYNOMIALS

INDEX1.INTRODUCTION

2. Different Types of polynomial

3.GEOMETRICAL MEANING OF ZEROES OF THE POLYNOMIAL

3.RELATION BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL

4.DIVISION ALGORITHM FOR POLYNOMIAL

What Is A Polynomial ???

A polynomial is an expression made with constants, variables and exponents, which are combined using addition, substraction and mutiplication but not division.

The exponents can only be 0,1,2,3…. etc.

A polynomial cannot have infinite number of terms.

For Example :-

2x2 +

3x =

5

x3 –

3x2 +

x +2

= 6

4y3 - 4y2 + 5y + 8

= 0

9x 2 + 9y + 8

=0

Different Types Of Polynomial

On the basis of number of terms—o Monomial – polynomials having only

one term.

e.g. 4x, 8y o Binomial – polynomials having two

terms. e.g. 2x + 6, 25y – 25

o Trinomial – polynomials having three terms.

e.g. 2x - x³ +25, x³ + 5x² -8

The exponent of

the highest

degree term in a

polynomial is

known as its

degree.

For example: f(x) = 3x + ½ is a

polynomial in the variable x of degree 1. g(y) = 2y2 – 3/2y + 7 is a polynomial in the variable y of degree 2. p(x) = 5x3 – 3x2 + x – 1/√2 is a polynomial in the

variable x of degree 3. q(u) = 9u5 – 2/3u4 + u2 –

½ is a polynomial in the variable u of degree 5.

Degree Of Polynomial

Constant Polynomial:

A polynomial

of degree

zero is

called a

constant

polynomial.

For example: f(x) = 7, g(x) = -3/2,

h(x) = 2are constant polynomials.

The degree of constant polynomials is not defined.

• A polynomial

of degree one

is called a

linear

polynomial.

For example:

p(x) = 4x – 3, q(x) = 3y are linear polynomials.

Any linear polynomial is in the form ax + b, where a, b are real

nos. and a ≠ 0.

It may be a monomial or a binomial. F(x) = 2x – 3 is binomial whereas

g (x) = 7x is monomial.

Linear Polynomial

Types of polynomial:Q U A D R A T I C

P O LY N O M I A L A polynomial of degree

two is called a quadratic polynomial.

f(x) = √3x2 – 4/3x + ½, q(w) = 2/3w2 + 4 are

quadratic polynomials with real coefficients.

Any quadratic is always in the form f(x) = ax2 + bx +c where a,b,c are

real nos. and a ≠ 0.

C U B I C P O L Y N O M I A L

A polynomial of degree three is called a cubic polynomial.

f(x) = 9/5x3 – 2x2 + 7/3x _1/5 is a cubic polynomial in variable x.

Any cubic polynomial is always in the form f(x = ax3 + bx2 +cx + d where a,b,c,d are real nos.

If f(x) is a polynomial and y is

any real no. then real no.

obtained by replacing x by y in

f(x) is called the value of f(x) at x

= y and is denoted by f(x).

Value of f(x) at x = 1

f(x) = 2x2 – 3x – 2

f(1) = 2(1)2 – 3 x 1 –

2

= 2 – 3 – 2

= -3

A real no. x is a zero of the

polynomial f(x),is f(x) = 0

Finding a zero of the polynomial

means solving polynomial equation f(x) = 0.Zero of the polynomial

f(x) = x2 + 7x +12 f(x) = 0 x2 + 7x + 12 = 0

(x + 4) (x + 3) = 0 x + 4 = 0 or, x + 3 = 0

x = -4 , -3

Value’s & zero’s of Polynomial

RELATIONSHIP B/W ZEROES AND COEFFICIENTS

OF A POLYNOMIAL

QUADRATIC

☻ α + β = - coefficient of

xCoefficient of x

2

= - b

a☻ αβ =

constant term

Coefficient of x2

= c

a

Cubic

• α + β + γ = -Coefficient

of x2 = -bCoefficient of x

3 = a

• αβ + βγ + γα = Coefficient

of x = c Coefficient of x3 = a

Relationships

ON VERYFYING THE

RELATIONSHIP BETWEEN

THE ZEROES AND

COEFFICIENTS

ON FINDING AN

UNKNOWN WHEN A

RELATION BETWEEEN

ZEROES AND

COEFFICIENTS ARE

GIVEN.

ON FINDING THE VALUES OF EXPRESSIONS

INVOLVING ZEROES OF QUADRATIC POLYNOMIAL

OF ITS A QUADRATIC POLYNOMIAL WHEN THE SUM AND PRODUCT OF ITS ZEROES ARE GIVEN.

DIVISION ALGORITHM

If f(x) and g(x) are any two

polynomials with g(x) ≠ 0,then we can always find

polynomials q(x), and r(x) such that :

F(x) = q(x) g(x) + r(x),

Where r(x) = 0 or degree r(x) < degree g(x)

ON VERYFYING THE DIVISION

ALGORITHM FOR POLYNOMIALS.

ON FINDING THE QUOTIENT AND

REMAINDER USING DIVISION

ALGORITHM.

ON CHECKING WHETHER A

GIVEN POLYNOMIAL IS A FACTOR OF THE

OTHER POLYNIMIAL BY

APPLYING THEDIVISION ALGORITHM

ON FINDING THE REMAINING

ZEROES OF A POLYNOMIAL WHEN

SOME OF ITS ZEROES ARE

GIVEN.