The number of polynomials having zeroes as - 2 and 5 is

let - 2 and 5 are the zeroes of the polynomials of the formp(x) = ax^{2} + bx + c.

sum of the zeroes = - (coefficient of x) ÷ coefficient of x^{2} 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 x^{2} 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) = ax^{2} + bx + c.

∴ The equation is x^{2} - 3x - 10

OR

The equation of a quadratic polynomial is given by x^{2} - (sum of the zeroes)x + (product of the zeroes) where,

Sum of the zeroes = - (coefficient of x) ÷ coefficient of x^{2} and

Product of the zeroes = constant term ÷ coefficient of x^{2}

∴ Sum of the zeroes = - 2 + 5 and product of the zeroes = (- 2)5

= 3 = - 10

∴ the equation is x^{2} - 3x - 10

We know, the zeroes do not change if the polynomial is divided or multiplied by a constant

⇒ kx^{2} - 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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