Question

If α and β are the zeroes of the polynomial ax² − x + c. Obtain a polynomial whose zeroes are α − 3 and β − 3.

Answer 1

α, β are zeroes of ax² − x + c.

α + β = `(-(-1))/a`

= `1/a`

α . β = `c/a`

The quadratic equation whose zeroes are α − 3 and β − 3 is

x² − [(α − 3) + (β − 3)]x + (α − 3)(β − 3) = 0

⇒ x² − [(α + β) − 6]x + αβ − 3(α + β) + 9 = 0

⇒ `x^2 - [1/a - 6]x + c/a - 3/a + 9 = 0`

⇒ `x^2 - ((1 - 6a)x)/a + (c - 3 + 9a)/a = 0`

⇒ ax² − (1 − 6a)x + (c − 3 + 9a) = 0

Required polynomial is ax² − (1 − 6a)x + (c − 3 + 9a).