Question

q 6,7,8 pls answer pls pls pls​

Answer 1

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.