Polynomials CLASS 10

10
POLYNOMIALS NAME – Nihas Kamarudheen CLASS- X - C ROLL NO-30 A presentation on

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POLYNOMIALS PPT FOR CLASS 10!! ENJOY

Transcript of Polynomials CLASS 10

Page 1: Polynomials CLASS 10

POLYNOMIALS

NAME – Nihas KamarudheenCLASS- X - CROLL NO-30

A presentation on

Page 2: Polynomials CLASS 10

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.

Page 3: Polynomials CLASS 10

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

Page 4: Polynomials CLASS 10

i) Consta

nt polynomial –

polnomials

having degree 0. e.g. 32, -5

ii) Linear p

olynomial – polynomials

having degree 1. e.g. x+5, 6x-3

ii) quadratic polynomial –

polynomials

having degree 2. e.g. 2

x² + 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. 2x 4 + x³ - 8x² +5x -8

On the basis of degree

Page 5: Polynomials CLASS 10

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.

Page 6: Polynomials CLASS 10

RELATIONSHIP BETWEEN THE ZEROES AND COEFFICIENTS OF A

QUADRATIC POLYNOMIAL

LET Α AND Β BE THE ZEROES OF THE POLYNOMIAL AX² + BX + C.

THEN,SUM OF ZEROES (Α + Β) = -B

A

= -(COEFFICIENT OF X)

COEFFICIENT OF X²

AND, PRODUCT OF ZEROES (ΑΒ) = C

A

= CONSTANT TERM

COEFFICIENT OF X²

Page 7: Polynomials CLASS 10

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

Page 8: Polynomials CLASS 10

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.

Page 9: Polynomials CLASS 10

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

Page 10: Polynomials CLASS 10

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

THE EN

D