Submitted By:-

Anshul Chauhan of Class xth

Polynomials

2x2 + 3x = 5

2x2 + 3x= 9

x 3 – 3x 2 + x +1 = 0 4y3 - 4y2 + 5y + 8 = 0

9x 2 + 9y + 8 =0

Introduction :

A polynomial is an expression of finite length constructed from

variables and constants, using only the operations of addition,

subtraction, multiplication, and non-negative, whole-number exponents.

Polynomials appear in a wide variety.

Let x be a variable 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.

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

Cont…

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:

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

are constant polynomials.The degree of constant polynomials is not defined.

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.

Types of polynomial:

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.

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.

Value’s & zero’s of Polynomial

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 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

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

GRAPHS O

F THE P

OLYNOMIAL

S

GRAPHS O

F THE P

OLYNOMIAL

S

GENERAL SHAPES OF POLYNOMIAL

f(x) = 3

CONSTANT FUNCTION

DEGREE = 0

MAX. ZEROES = 0

1

Cont….

f(x) = x + 2

LINEAR FUNCTION

DEGREE =1

MAX. ZEROES = 1

2

Cont…

f(x) = x2 + 3x + 2

QUADRATIC FUNCTION

DEGREE = 2

MAX. ZEROES = 2

3

Cont…

f(x) = x3 + 4x2 + 2

CUBIC FUNCTION

DEGREE = 3

MAX. ZEROES = 3

4

QUADRATICQUADRATIC

☻ α + β = - coefficient of x

Coefficient of x2

= - ba

☻ αβ = constant termCoefficient of x2

= ca

CUBICCUBIC

α + β + γ = -Coefficient of x2 = -bCoefficient of x3 a

αβ + βγ + γα = Coefficient of x = cCoefficient of x3 a

αβγ = - Constant term = dCoefficient of x3 a

Relationships

ON VERYFYI

NG THE

RELATIONSHIP B

ETWEEN

THE ZEROES A

ND

COEFFICIEN

TS

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.

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),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.

THANKS FOR BEING PATIENT