Question

Find a quadratic polynomial whose sum and product respectively of the zeros are given. Also, find the zeros of these polynomials.

`-2sqrt(3), -9`

Answer 1

Given: Sum of zeros = `-2sqrt(3)`, Product of zeros = –9.

Step-wise calculation:

1. A monic quadratic with those sum (S) and product (P) is x² – Sx + P.

2. Substitute `S = -2sqrt(3)` and P = –9:

`x^2 - (-2sqrt(3))x + (-9) = x^2 + 2sqrt(3)x - 9`

3. Factorise: `x^2 + 2sqrt(3)x - 9`

= `x^2 + 3sqrt(3)x - sqrt(3)x - 9` 

= `(x + 3sqrt(3))(x - sqrt(3))`

Required quadratic polynomial: `x^2 + 2sqrt(3)x - 9` 

Zeros: `x = sqrt(3)` and `x = -3sqrt(3)`.