Polynomials CLASS 10
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Transcript of Polynomials CLASS 10
POLYNOMIALS
NAME – Nihas KamarudheenCLASS- X - CROLL NO-30
A presentation on
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.
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
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
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.
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
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
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
THE EN
D