Question

If alpha and beta are the zeros of the quadratic polynomial x square +px+q form a polynomial whose zeros are (alpha+beta) square and (alpha-beta) square.

Answer 1

let 2 zeroes be a and b of  polynomial  x² + px + q = 0

sum of roots = a + b = -p/1 = -p
products of roots = ab = q/1 = q

(a + b)² = a² + b² + 2ab ⇒  a² + b² = p² - 2q   

(a-b)² = a² + b² -2ab =  p² - 2q -2q = p² - 4q   



now ques asks for new quadratic eq whose roots are (a+b) ²  and  (a-b)²

so sum of new roots are = (a +b)² + (a-b)² = p² + p² -4q = 2p² - 4q

and  product of roots =  (a+b)²(a-b)² =  (p²)² (p²-4q)² = p⁴ (p⁴ +16q² + 8p²q)

hence new quadratic eq gonna be =

   x² - x(sum of roots) + (products of roots)

  x² - x(2p² - 4q) + p⁴(p⁴ + 16q² +8p²q) = 0

further  u can simplify.


hope my ans is correct ...........doubts , then enquire.

Answer 2

We have polynomial  f ( x ) =  x2  + p x  + q
And
Roots are α  and β
And
we know from relationship between zeros and coefficient . 

Sum of zeros  = −Coefficient of xCoefficient of x2
So,
α  + β  =  - p                                         ------ ( 1 )

Taking whole square on both hand side , we get

( α  + β  )2 = p 2                                  ------ ( 2 )

⇒α2 + β2 +  2 α β = p2⇒α2 + β2 +  2 α β − 2αβ + 2αβ=  p2⇒α2 + β2 −  2 α β  +4 αβ=  p2⇒(α  − β) 2  +4 αβ=  p2                     −−−− ( 3 )

And

Products of zeros  = Constant termCoefficient of x2
So,
α  β  =  q        , Substitute that value in equation 3 , we get

⇒(α− β)2 +  4 (q ) = p2⇒(α− β)2 +4 q=  p2⇒(α− β)2  = p2 − 4 q              −−−− ( 4 )   

Now we add equation 2 and 4 and get

(α + β)2 + (α − β)2 = p2 +  p2 − 4 q= 2 p2 − 4 q

And we multiply equation 2 and 4 and get

(α + β)2 × (α − β)2 = p2( p2 − 4 q)= p4 − 4 p2q

And we know formula for polynomial when sum of zeros and product of zeros we know :

Polynomial  =  k [ x2  - ( Sum of zeros ) x  + ( Product of zeros ) ]   , Here k is any non zero real number.

Substitute values , we get

Quadratic polynomial  =  k [ x2  - ( 2 p2 - 4 q) x  + ( 2 p4 - 4 p2q) ] 

                             
= x2  - ( 2 p2 - 4 q) x  + ( 2 p4 - 4 p2q) [ taking k = 1 ]                                         ( Ans )

Hope this information will clear your doubts about topic.