Question

Find the quadratic polynomial whose zeros are 2 and –6. Verify the relation between the coefficients and the zeros of the polynomial.

Answer 1

Given: Zeros α = 2 and β = –6.

Step-wise calculation:

1. Sum of zeros:

α + β = 2 + (–6) 

= –4

2. Product of zeros:

αβ = 2 × (–6) 

= –12

3. A monic quadratic with these zeros is

x² – (α + β)x + αβ 

= x² – (–4)x + (–12) 

= x² + 4x – 12

The required quadratic polynomial is x² + 4x – 12. For this polynomial (a = 1, b = 4, c = –12), `−b/a = -4` equals α + β and `c/a = -12` equals αβ, so the relation between coefficients and zeros is verified.