Presented By;-NAME – Ashish Pradhan , Durgesh KumarCLASS- X – ‘A’ROLL NO-27 , 26

A presentation on

INTRODUCTIONGEOMETRICAL MEANING OF ZEROES OF THE POLYNOMIAL

RELATION BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL

DIVISION ALGORITHM FOR POLYNOMIAL

Polynomials are algebraic expressions that include real numbers and variables. The power of the variables should always be a whole number. Division and square roots cannot be involved in the variables. The variables can only include addition, subtraction and multiplication.Polynomials contain more than one term. Polynomials are the sums of monomials. A monomial has one term: 5y or -8x2 or 3. A binomial has two terms: -3x2 2, or 9y - 2y2

A trinomial has 3 terms: -3x2 2 3x, or 9y - 2y2 y The degree of the term is the exponent of the variable: 3x2 has a

degree of 2.When the variable does not have an exponent - always understand that there's a '1' e.g., 1x

Example:x2 - 7x - 6 (Each part is a term and x2 is referred to as the leading term)

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.

Let “x” be a variable and “n” be a positive integer and as, a1,a2,….an be constants (real nos.)

Then, f(x) = anxn+ an-1xn-1+….+a1x+xo

anxn,an-1xn-1,….a1x and ao are known as the terms of the polynomial.

an,an-1,an-2,….a1 and ao are their coefficients.

For example:• p(x) = 3x – 2 is a polynomial in variable x.• q(x) = 3y2 – 2y + 4 is a polynomial in variable y.• f(u) = 1/2u3 – 3u2 + 2u – 4 is a polynomial in variable u.

NOTE: 2x2 – 3√x + 5, 1/x2 – 2x +5 , 2x3 – 3/x +4 are not polynomials.

DIFFERENT TYPES OF

POLYNOMIALS

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 degree is the term with the greatest exponent

Recall that for y2, y is the base and 2 is the exponent

For example: p(x) = 10x4 + ½ is a polynomial in the variable x of degree 4.

p(x) = 8x3 + 7 is a polynomial in the variable x of degree 3.

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

p(x) = 8u5 + u2 – 3/4 is a polynomial in the variable x of degree 5.

DEGREE

i) Constant polynomial – polnomials having degree 0.

e.g. 32, -5

ii) Linear polynomial – polynomials having degree 1.

e.g. x+5, 6x-3

ii) quadratic polynomial – polynomials having degree 2.

e.g. 2x² + 3x -8

iii) Cubic polynomial – polynomials having degree 3.

e.g. 6x³ + 7x² -x-6

v) bi-quadratic polynomial- polynomials having degree 4.

e.g. 2x4 + x³ - 8x² +5x -8

More information of degree

ZEROES OF A POLYNOMIAL

A real number α is a zero of a

polynomial f(x), if f(α) = 0.

e.g. f(x) = x³ - 6x² +11x -6

f(2) = 2³ -6 X 2² +11 X 2 – 6

= 0 .Hence 2 is a zero

of f(x).

The number of zeroes of the

polynomial is the degree of the polynomial. Therefore a quadratic

polynomial has 2 zeroes and cubic

3 zeroes.

For example: f(x) = 7, g(x) = -3/2, h(x) = 2are constant polynomials. The degree of constant polynomials is

ZERO.

For example: p(x) = 4x – 3, p(y) = 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.

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 polynomial is always in the form:-

ax2 + bx +c where a,b,c are real nos. and a ≠ 0.

• A polynomial of degree three is called a cubic

polynomial.• f(x) = 5x3 – 2x2 + 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.

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.

If p(x) is a polynomial and “y” is any real no. then real no. obtained by replacing “x” by “y”in p(x) is called the value of p(x) at x = y and is denoted by “p(y)”.

For example:-Value of p(x) at x = 1 p(x) = 2x2 – 3x – 2 p(1) = 2(1)2 – 3 x 1 – 2 = 2 – 3 – 2 = -3

For example:-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 = 0x = -4 , -3

VALUE OF POLYNOMIAL

ZERO OF A POLYNOMIAL

QUADRATIC

☻ A + B = - Coefficient of x

Coefficient of x2

= - ba

☻ AB = Constant term Coefficient of x2

= ca

Note:- “A” and “B” are the zeroes.

RELATIONSHIP BETWEEN THE ZEROES AND COEFFICIENTS OF A

QUADRATIC POLYNOMIAL

An nth degree polynomial can have at most “n” real zeroes.

Graphs of the polynomialsNumber of real zeroes of a polynomial is less than or equal to degree of the polynomial.

GENERAL SHAPES OF POLYNOMIAL

FUNCTIONS f(x) = x + 2

LINEAR FUNCTION

DEGREE =1

MAX. ZEROES = 1

GENERAL SHAPES OF POLYNOMIAL FUNCTIONS

f(x) = x2 + 3x + 2

QUADRATIC FUNCTION

DEGREE = 2

MAX. ZEROES = 2

Relationship between the zeroes and coefficients of a cubic polynomial

• Let α, β and γ be the zeroes of the polynomial ax³ + bx² + cx + d

• Then, sum of zeroes(α+β+γ) = -b = -(coefficient of x²)

a coefficient of x³

αβ + βγ + αγ = c = coefficient of x

a coefficient of x³

Product of zeroes (αβγ) = -d = -(constant term)

a coefficient of x³

GENERAL SHAPES OF POLYNOMIAL FUNCTIONS

f(x) = x3 + 4x2 + 2

CUBIC FUNCTION

DEGREE = 3

MAX. ZEROES = 3

ON VERYFYING THE

RELATIONSHIP BETWEEN

THE ZEROES AND

COEFFICIENTS

ON FINDING THE

VALUES OF EXPRESSIONS

INVOLVING ZEROES OF

QUADRATIC POLYNOMIAL

ON FINDING AN UNKNOWN

WHEN A RELATION

BETWEEEN ZEROES AND

COEFFICIENTS ARE GIVEN.

OF ITS A QUADRATIC

POLYNOMIAL WHEN

THE SUM AND

PRODUCT OF ITS

ZEROES ARE GIVEN.

RELATIONSHIPS.

DIVISION ALGORITHM FOR POLYMIALS

If p(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 :

P(x) = q(x) g(x) + r(x),Where r(x) = 0 or degree r(x) < degree g(x)

QUESTIONS BASED ON POLYNOMIALS

I) Find the zeroes of the polynomial x² + 7x + 12and verify the relation between the zeroes and its coefficients.

f(x) = x² + 7x + 12

= x² + 4x + 3x + 12

=x(x +4) + 3(x + 4)

=(x + 4)(x + 3)

Therefore,zeroes of f(x) =x + 4 = 0, x +3 = 0 [ f(x) = 0]

x = -4, x = -3

Hence zeroes of f(x) are α = -4 and β = -3.

Sum of zeroes = α + β = -4 -3 = -7 -(coefficient of x) = -7 coefficient of x²Hence, sum of zeroes = -(coefficient of x) coefficient of x²Product of zeroes = αβ = (-4)(-3) = 12Constant term = 12Coefficient of x²Hence, product of zeroes = constant term coefficient of x²

2) Find a quadratic polynomial whose zeroes are 4, 1.

sum of zeroes,α + β = 4 +1 = 5 = -b/a

product of zeroes, αβ = 4 x 1 = 4 = c/a

therefore, a = 1, b = -4, c =1

as, polynomial = ax² + bx +c

= 1(x)² + { -4(x)} + 1

= x² - 4x + 1

2) Find a quadratic polynomial whose zeroes are 4, 1.

sum of zeroes,α + β = 4 +1 = 5 = -b/a

product of zeroes, αβ = 4 x 1 = 4 = c/a

therefore, a = 1, b = -4, c =1

THE END