Question

the zeros of the polynomial f(x) =9x^2-18+8 are alpha and beta. find a quadratic polynomial having zeros alpha squre and beta square​

Answer 1

Step-by-step explanation:

In a quadratic equation, ax^2 + bx + c,   sum S of roots is given by - b/a   and product P of roots is c/a.

In this equation,

S = α + β = -(-18/9) = 2

P = αβ = 8/9

       Hence, (αβ)^2 = (8/9)^2

                               = 64/81

Also,

⇒ (α + β)^2 = 2^2

⇒ α^2 + β^2 + 2αβ = 4

⇒ α^2 + β^2 + 2(8/9) = 4

⇒ α^2 + β^2 = 20/9

 So, if we form an equation with roots α^2 and β^2,

sum of roots = α^2 + β^2 = 20/9

product of roots = (αβ)^2 = 64/81

  Hence, equation is:

⇒ x^2 - (20/9)x + (64/81)

⇒ (81x^2 - 180x + 64)/81

  Or, we directly say 81x^2 - 180x + 64