Question

if alpha and beta are zeros of the polynomial x^2+7x+3, find a quadratic polynomial whose zeroes are alpha/beta and beta/alpha​

Answer 1

Quadratic equation,

3x²-43x+3 = 0

f(x) = x²+7x+3. (1)

quadratic equations are in the form of,

x²-(α+β)x+αβ = 0

frm(1), α+β = -7

αβ = 3

If the zeroes of the polynomial is

α/β and β/α,

x²-[(α²+β²)/αβ]x+ 1 = 0

α/β + β/α = (α²+β²) / αβ

= ((α+β)²-2αβ) / αβ

= (49-6) / 3

= 43 / 3

x²-[(α²+β²)/αβ]x+ 1 = 0

x²-43x/3+1 = 0

(×3) 3x²-43x+3 = 0