Question

Find the quadratic polynomial, the sum of whose zeros is 0 and their product is -1. Hence, find the zeros of the polynomial.

Answer 1

Let α, β be the zeros of required quadratic polynomial f(x)

We know that ${(\alpha +\beta )}^2$ = $x^2$ + $Sumoftheroots\times x$ + product of roots

We have,

α+β = 0, αβ= -1

Therefore, Polynomial whose zeros are α, β is

= $x^2$ - (α+β)x + αβ

= $x^2$ – 0.x + (-1) = $x^2$ – 1

Therefore, Required polynomial is $x^2$– 1

Now f(x) = $x^2$ – 1

$\to$ = (x – 1)(x + 1)

f(x) = 0

$\to$ (x – 1)(x + 1) = 0

Therefore, Either x – 1 = 0 or x + 1 = 0

i.e., Either x = 1 or x = -1

Therefore, Zeros of the polynomial are 1 and -1