Question

Question: 19
If α and β are the zeroes of the quadratic polynomial f(x) = x2 + px + q, form a polynomial whose zeroes are (α + β)2 and (α – β)2.
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Answer 1

Solution:

We have,

f(x) = x2 + px + q

Sum of the zeroes = α + β = -p

Product of the zeroes = αβ = q

From the question,

Sum of the zeroes of new polynomial = (α + β)2 + (α – β)2

= (α + β)2 + α2 + β2 – 2αβ

= (α + β)2 + (α + β)2 – 2αβ – 2αβ

= (- p)2 + (- p)2 – 2 × q – 2 × q

= p2 + p2 – 4q

= p2 – 4q

Product of the zeroes of new polynomial = (α + β)2 (α – β)2

= (- p)2((- p)2 - 4q)

= p2 (p2–4q)

So, the quadratic polynomial is,

x2 – (sum of the zeroes)x + (product of the zeroes)

= x2 – (2p2 – 4q)x + p2(p2 – 4q)

Hence, the required quadratic polynomial is f(x) = k(x2 – (2p2 –4q) x + p2(p2 - 4q)).

Answer 2

Answer:

⤵⤵Your Answer⤵⤵

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F(x) =2x + Px +Q

Sum of the zeroes = A + B = -P

Then , = AB = Q

Now ,

Polynomial ( A + B ) 2 + ( A - B ) 2

= ( A + B ) 2 + A2 + B2 - 2AB

= ( A + B ) 2 + ( A + B ) 2 - 2AB - 2AB

= ( - P ) 2 + ( - P ) 2 - 2 × Q - 2 × Q

= P2 + P2 - 4Q

= P2 - 4Q

Now : Polynomial

= ( A + B ) 2 ( A - B ) 2

= (- P )2 (- P )2 - 4Q

= P2 ( P2 - 4Q )

So , Polynomial :

On the place of zero = X2

= X2 - ( 2Q 2 - 4Q ) X + P2 ( P2 - 4Q )

Polynomial : ( x ) = K( x2 ) - (2P^2 - 4Q ) x + P2 ( P2 - 4Q )

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