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.
6.
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.
9.
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
