Question

If alpha and beta are the zeros of the quadratic polynomial f(x)=x^2-3x-2 find a quadratic polynomial whose zeros are 1/2 alpha+beta and 1/2 beta+alpha

Answer 1

Answer:

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Answer 2

Answer:

Answer:

x^2 + 13x + 4

Step-by-step explanation:

 Polynomials written in the form of x^2 - Sx + P represent S as sum of their roots and P as product of their roots.

  Here, if α and β are roots.

 S = α + β = 3    

 P = αβ = - 2

     From above,

⇒ α + β = 3

⇒ ( α + β )^2 = 3^2

⇒ α^2 + β^2 + 2αβ = 9

             αβ = - 2

⇒ α^2 + β^2 + 2( - 2 ) = 9

⇒ α^2 + β^2 - 4 = 9

⇒ α^2 + β^2 = 9 + 4 = 13

      For the other equ.

Sum of its roots = 2α/β + 2β/α

     = 2[ α/β + β/α ]

    = 2[ α^2 + β^2 ]/αβ

    = 2[ 13 / ( - 2 ) ]

    = - 13

Product of roots = 2(α/β)*2(β/α)

           = 2 * 2

           = 4

Hence the required polynomial is x^2 - ( - 13 )x + 4  ⇒ x^2 + 13x + 4.