This document contains a presentation on polynomials. It defines what a polynomial is and discusses how the degree of a polynomial relates to the number of zeros. It provides an example of a cubic polynomial and discusses the relationship between the zeros and coefficients of a cubic polynomial. Specifically, it states that the sum of the zeros equals the negative of the coefficient of x^2, the sum of the products of the zeros equals the coefficient of x, and the product of the zeros equals the negative of the constant term. The document then provides two example questions - one asking to find the zeros of a quadratic polynomial and verify the relationship between the zeros and coefficients, and another asking to find a quadratic polynomial with given zeros.

1. NAME – Nihas Kamarudheen
CLASS- X - C
ROLL NO-30
A
presentation
on

2. WHAT IS A POLYNOMIAL

4. On the basis of degree

5. 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.

7. Relationship between the zeroes and coefficients of a cubic
polynomial
• Let α, β and γ be the zeroes of the polynomial ax³ + bx² + cx
• 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

8. 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.

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