Question

Find the quadratic polynomial, the sum of whose zeroes is `(5/2)` and their product is 1. Hence, find the zeros of the polynomial.

Answer 1

Let α and β be the zeroes of the required polynomial f(x).  

Then (α + β) = `5/2` and αβ = 1 

∴ f(x) = x² – (α + β)x + αβ 

⇒ f(x) = `x^2 - 5/2 x + 1`

⇒ f(x) = 2x² – 5x + 2

Hence, the required polynomial is f(x) = 2x² – 5x + 2

∴ f(x) = 0 ⇒ 2x² – 5x + 2 = 0

⇒ 2x² – (4x + x) + 2 = 0

⇒ 2x² – 4x – x + 2 = 0

⇒ 2x(x – 2) – 1(x – 2) = 0

⇒ (2x – 1) (x – 2) = 0

⇒ (2x – 1) = 0 or (x – 2) = 0

⇒ x = `1/2` or x = 2 

So, the zeros of f(x) are `1/2` and 2.