Q. 44.0( 44 Votes )
The number of polynomials having zeroes as - 2 and 5 is
D. more than 3
let - 2 and 5 are the zeroes of the polynomials of the form p(x) = ax2 + bx + c
Sum of the zeroes = -(coefficient of x) ÷ coefficient of x2 i.e.
Sum of the zeroes = - b/a
- 2 + 5 = - b/a
3 = - b/a
⇒ b = - 3 and a = 1
Product of the zeroes = constant term ÷ coefficient of x2 i.e.
Product of zeroes = c/a
(- 2)5 = c/a
- 10 = c
Put the values of a, b and c in the polynomial p(x) = ax2 + bx + c
∴ The equation is x2 - 3x - 10
The equation of a quadratic polynomial is given by x2 - (sum of the zeroes)x + (product of the zeroes) where,
Sum of the zeroes = - (coefficient of x) ÷ coefficient of x2 and
Product of the zeroes = constant term ÷ coefficient of x2
∴ Sum of the zeroes = - 2 + 5 and product of the zeroes = (- 2)5
= 3 = - 10
∴ the equation is x2 - 3x - 10
We know, the zeroes do not change if the polynomial is divided or multiplied by a constant
⇒ kx2 - 3kx - 10k will also have - 2 and 5 as their zeroes.
As k can take any real value, there can be many polynomials having - 2 and 5 as their zeroes.
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