POWER POINT PRESENTATION

FOR

CLASS X MATHEMATICS

SATHEESH KUMAR K

TGT, AECS-5,MUMBAI

CHAPTER TWO

POLYNOMIALS

TOPICS

Introduction

Geometrical Meaning of the Zeros of a Polynomial

Relationship Between Zeros and Coefficients of a Polynomial

Division Algorithm for Polynomials

2.1 INTRODUCTION

Definition : An algebraic

expression is an expression

built up from constants,

variables and the algebraic

operations(addition, subtraction

multiplication and division)

for example :

x+3, 3y-8, 7x², 5xy+8z-√3z³ etc

Consider the expression 5x³ - 4xyz + 8In this expression 5x³, - 4xyz and 8 are theterms.Terms are added to form expressions.Terms themselves are formed as product offactors.In general, any expression containing one ormore terms with non zero coefficients (andvariables with non negative integers asexponents) is called a polynomial.

TYPES OF POLYNOMIAL(NO. OF TERMS)

A polynomial of one term is called a

monomial.

Examples: 2x, 3xyz, -5, ¾ z etc

A polynomial of two terms is called a

binomial.

Examples: 5y-3, 4z³+7, 5xyz –x etc

A polynomial of three terms is called. a

trinomial

Examples:90xz+16x -¼, x-y-7,

2ax +3by –xy etc

DEGREE OF A POLYNOMIAL

The highest power of the variable in a polynomial

is the degree of the polynomial.

So, the degree of the polynomial 3x7+ 6x4- 4x +8

is 7 and the degree of the polynomial

5y6 + 9y3 – 10 is 6.

The degree of a non-zero constant polynomial

is zero.

TYPES OF POLYNOMIALS (DEGREE)

A polynomial of degree one is called a linear

polynomial.

Examples: 3x-5, 8x+7y-9z, ½ x-6z-10√2 etc

A polynomial of degree two is called a quadratic

polynomial.

Examples: 3x²-5x+4, 8xy+7y-9z, ½ x-6z²-Π etc

A polynomial of degree three is called a cubic

polynomial

Examples: x³+13x²-5x+14, 8xy+7y-9z³,

x-6z²-8y³

POLYNOMIALS IN ONE VARIABLE

A polynomial p(x) in one variable x is an

algebraic expression in x of the form

p(x) = anxn + an-1xn-1 + . . . + a2x2 + ax + a0,

where a0, a1, a2, . . ., an are constants(real

numbers) and an ≠ 0.

a0, a1, a2, . . ., an are respectively the coefficients

of x0, x, x2, . . ., xn, and n is called the degree of

the polynomial. Each of anxn , an-1xn-1 , . . . ,a2x2 ,

ax , a0is called a term of the polynomial p(x).

In particular,

if a0= a1 = a2 = . . . = an = 0

(all the constants are zero), we get

the zero polynomial, which is

denoted by 0.

The degree of the zero

polynomial is not defined.

ZERO OF A POLYNOMIAL

A real number ‘a’ is a zero of a polynomial p(x) if

p(a) = 0. In this case, a is also called a root of the

equation p(x) = 0.

Every linear polynomial in one variable has a

unique zero, a non-zero constant polynomial

has no zero, and every real number is a zero of

the zero polynomial.

A quadratic polynomial can have at most 2

zeroes and a cubic polynomial can have at

most 3 zeroes

2.2 Geometrical Meaning of the Zeroes of a

Polynomial

The linear polynomial ax + b, a ≠ 0, has exactly one

zero, namely –b/a the x-coordinate of the point where

the graph of y = ax + b intersects the x-axis.

.

Geometrical Meaning of the Zeroes of a Polynomial

Example : The zero of the linear polynomial

-2x +5 is 5/2 the point where the graph of the

linear equation y = -2x+ 5 meets the x axis.

Please refer the following graph.

GRAPH OF LINEAR EQUATION Y= -2X + 5

For any quadratic polynomial ax² + bx + c, a ≠ 0,

the graph of the corresponding equation

y = ax² + bx + c has one of the two shapes U either

open upwards or open downwards depending on

whether a > 0 or a < 0.

These curves are called parabolas.

A parabola is a plane curve which is mirror

symmetrical and approximately U-shaped.

Please refer the foll.owing figure

The zeroes of a quadratic

polynomial ax² + bx + c, a ≠ 0, are

precisely the x-coordinates of the

points where the parabola

representing y = ax² + bx + c

intersects the x-axis

We can see geometrically, from the following graphs,

that a quadratic polynomial can have either two

distinct zeroes or two equal zeroes (i.e., one zero), or no

zero. This also means that a polynomial of degree 2

has at most two zeroes

Fig- 1

In general, given a

polynomial p(x) of degree n,

the graph of y = p(x)

intersects the x-axis at at

most n points. Therefore, a

polynomial p(x) of degree n

has at most n zeroes.

2.3 RELATIONSHIP BETWEEN ZEROES AND

COEFFICIENTS OF A POLYNOMIAL

Consider a quadratic polynomial,

say p(x) = 2x² – 7x + 6.

Factorise quadratic polynomials by splitting the

middle term.

2x²– 7x + 6 = 2x² – 4x – 3x + 6 = 2x(x – 2) –3(x – 2)

= (x – 2)(2x – 3)

= (x – 2)(2x – 3)

So, the value of p(x) = 2x² – x + 6 is zero

when x –2 = 0 or 2 x – 3 = 0, i.e., when

x = 2or x = 3/2. So, the zeroes of 2x² – 7x + 6 are

2 and 3/2.

Observe that :

Sum of its zeroes = 2 +3/2 =7/2

= − (coefficient of x)/(coefficient ofx²)

Product of its zeroes = 2 X3/2 = 3 = 6/2

= (Constant term)/(coefficient of x²)

In general, if α and β are the zeroes of the quadratic

polynomial

p(x) = ax² + bx + c, a≠0 then x – α and x –β are the

factors of p(x). Therefore,

ax² + bx + c = k(x –α ) (x –β ), where k is a constant

= k[x² – (α + β )x + α β

= kx² – k(α + β )x + k α β

Comparing the coefficients of x², x and constant terms on

both the sides, we get

a = k, b = – k(α + β ) and c = kα β

This gives,

α + β = - b/a

αβ = c/a

In general, if α and β are the zeroes of

the quadratic polynomial

p(x) = ax² + bx + c, a≠0 then,

α + β = −b/a

αβ = c/a

2.4 DIVISION ALGORITHM FOR

POLYNOMIALS

If p(x) and g(x) are any two polynomials with

g(x) ≠0, then we can find

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

p(x) = g(x) × q(x) + r(x),

where r(x) = 0 or degree of r(x) < degree of g(x).

This result is known as the Division Algorithm

for polynomials

Example: Divide.x³ – 4x² + 2 x -3 by x+2

Solution :.

So, here the quotient is x² – 6x + 14 and

the remainder is − 31. Also,

(x+2)(x² –6x+ 14) + (−31 )

=x³− 6x² +14x +2x² −12x +28 −31

= x³ −4x² + 2x -3

Therefore,

Dividend = Divisor × Quotient + Remainder

If p(x) and g(x) are any two polynomials with

g(x) ≠ 0, then we can find

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

p(x) = g(x) × q(x) + r(x),

where r(x) = 0 or degree of r(x) < degree of g(x).

This result is known as the Division

algorithm for polynomials.

QUESTIONS FOR PRACTISE:

1)Find the zeros of the polynomial p(x) =3x² - x -4

and verify the relationship between zeros and

coefficients.

2) Divide x³ -3x² +3x + 5 by x -2 and verify

division algorithm for polynomials.

3) Questions from Exercise 2.1,2.2 and 2.3 of

Class X NCERT Mathematics Text book

Thank you