Question

If α and β are the zeros of the quadratic polynomial f(x) = x² + px + q , from a polynomial whose zeros are (α + β)² (α - β)² .

Answer 1

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Answer 2

f(x) = k(x² -2(p²-q)x +p^4 -4p²q)

Step-by-step explanation:

  f(x) = x² + px + q

α and β are the zeroes of the polynominial.

Then α + β = -b/a = -p

αβ = c/a  = q

 

(α + β)² and (α - β)²  are the zeroes.

Then the polynomial will be f(x) = k(x² -Sx +P)

S =  (α + β)² + (α - β)²  = (α² + β² + 2αβ) + (α² + β² - 2αβ)  

 = 2(α² + β²)

 = 2 [(α + β)²-αβ]

 = 2 [ (-p)² - q ]

 = 2p²-2q

S  = 2(p²-q)

P = (α + β)²*(α - β)²  

 = (-p)² * (α² + β² - 2αβ)  

 = p² *[ (α + β)² - 2αβ - 2αβ)]

 =  p² *[ (-p)² - 2q - 2q]

 =  p²(p² - 4q)

P  = p^4 -4p²q

So the polynomial will be:

f(x) = k(x² -2(p²-q)x +p^4 -4p²q)