Question

If alpha and beta are zeros of the polynomial 2x^2-4x+5 then form a polynomial whose zeros are 1/alpha^2 + 1/beta^2

Answer 1

f(x) = 2x^2-4x+5
(Here a=2,b= -4 & c =5)
Zeros- alpha and beta
Sum of the zeros = -b/a —> alpha + beta = -( -4/2 ) = 2–> (1)
Product of the zeros = c/a —> (alpha)(beta) = 5/2 —> (2)

Now for the new quadratic polynomial, the zeros are 1/alpha^2 and 1/beta^2,
Sum of these zeros = 1/alpha^2 + 1/beta^2 =[ beta^2 + alpha^2]/[(alpha)(beta)]^2
=[( alpha+beta)^2 - 2(alpha)(beta)]/[(alpha)(beta)]^2
= [(2)^2 - 2(5/2)]/(5/2)^2. [From (1) and (2)]
= -4/25

Similarly, product of the zeros = (1/alpha^2)(1/beta^2) = 1/[(alpha)(beta)]^2
= 1/(5/2)^2 = 4/25
Let the new polynomial be of the form x^2-x(sum) + (product) = 0
—> k(x^2 + 4x/25+ 4/25) = 0