Alpha +2/beta+2+beta+2/alpha+2
Answers
Answer:
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.