Question

If alpha and beta are the are zeros of polynomial x^{2} -2x+5 then form a quadratic polynomial whose zeros are alpha+beta and 1/alpha+1/beta.

Answer 1

Given that α and β are the zeroes of the polynomial, x²-2x+5,

We can say,

α+β=-b/a=2 ---------(i)

and, αβ=c/a=5 -----------(ii)

Now, for a polynomial to have zeroes (α+β) and (1/α+1/β),

f(x)=x²-(α+β+1/α+1/β)x+(α+β)(1/α+1/β)

=x²-{α+β+(α+β)/αβ}x+(α+β){(α+β)/αβ)}

Putting the values from (i) and (ii),

=x²-{2+(2/5)}x+2(2/5)

=x²-(12/5)x+4/5

=k[5x²-12x+4] (Ans.)                       (k is any non-zero constant)